{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0001","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2326818b52a7c129bf8a44e36c0afc9de8a6ffad590686272556e53a5786274","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156001,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [[v0, v2], [v1, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [[v0, v2], [v1, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0002","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90febc8c5efa440a73915c54b5f248042282201a4945b8fe1452f64a5ca081e8","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156002,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [[v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 2, [[v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [[v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 2, [[v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0003","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be46408cc3adc3972d16e03e27104c8ad46cdf3b7d20843b571998a1e740761","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156003,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v2, v4], [v1, v3, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v2, v4], [v1, v3, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0004","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ad074b0b7959787deb0264891a329515999536691560764b60a48ba03fbc679","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156004,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 3, [[v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 3, [[v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0005","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78543c2a5f5e5f4abd0df683a34021ed274e9d78cbfb1be3b20ee3b13b72c594","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156005,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0006","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c01d372eaf1a7c3c620b1115a614df5d52c2a88aa0ca1592e9be1eea7a2c0e6c","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156006,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 4, [[v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 4, [[v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0007","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c7d0bfb9e04f44d86ff1556ae5454e53d4b321c4b0b0e03587bb1d3937b5b01","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156007,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0008","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e01ff3d51cb98131cd3f602b3809ff533afd8d7dc0eeb002280b611f4b55a890","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156008,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0009","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69375ef853ef31c296fdbb304791ebcd5a7e6708bfc9c907ce6300533cd376ad","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156009,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v1, v10, v11, v3, v5, v6, v8], [v0, v1, v10, v11, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, 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v8, v9], [v0, v1, v11, v2, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v1, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v1, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v1, v10, v11, v3, v5, v6, v8], [v0, v1, v10, v11, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v11, v2, v3, v5, v6, v8], [v1, v10, v11, v2, v3, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v0, v10, v11, v2, v4, v5, v6, v8], [v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v1, v10, v11, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v1, v10, v11, v3, v5, v7, v8], [v0, v1, v10, v11, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v11, v2, v3, v5, v7, v8], [v1, v10, v11, v2, v3, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v0, v10, v11, v2, v4, v5, v7, v8], [v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v1, v10, v11, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v0, v10, v11, v2, v4, v6, v7, v8], [v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v1, v10, v11, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v1, v10, v11, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9], [v0, v1, v11, v3, v5, v7, v9], [v1, v10, v11, v3, v5, v7, v9], [v0, v1, v10, v11, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v11, v2, v3, v5, v7, v9], [v1, v11, v2, v3, v5, v7, v9], [v0, v1, v11, v2, v3, v5, v7, v9], [v0, v10, v11, v2, v3, v5, v7, v9], [v1, v10, v11, v2, v3, v5, v7, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v0, v11, v2, v4, v5, v7, v9], [v1, v11, v2, v4, v5, v7, v9], [v0, v1, v11, v2, v4, v5, v7, v9], [v0, v10, v11, v2, v4, v5, v7, v9], [v1, v10, v11, v2, v4, v5, v7, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v1, v11, v3, v4, v5, v7, v9], [v0, v1, v11, v3, v4, v5, v7, v9], [v1, v10, v11, v3, v4, v5, v7, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v11, v2, v3, v4, v5, v7, v9], [v1, v11, v2, v3, v4, v5, v7, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v0, v11, v2, v4, v6, v7, v9], [v1, v11, v2, v4, v6, v7, v9], [v0, v1, v11, v2, v4, v6, v7, v9], [v0, v10, v11, v2, v4, v6, v7, v9], [v1, v10, v11, v2, v4, v6, v7, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v1, v11, v3, v4, v6, v7, v9], [v0, v1, v11, v3, v4, v6, v7, v9], [v1, v10, v11, v3, v4, v6, 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[v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v11, v2, v3, v5, v6, v8, v9], [v1, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v11, v2, v4, v5, v6, v8, v9], [v1, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v1, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v1, v11, v3, v5, v7, v8, v9], [v0, v1, v11, v3, v5, v7, v8, v9], [v1, v10, v11, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v11, v2, v3, v5, v7, v8, v9], [v1, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v11, v2, v4, v5, v7, v8, v9], [v1, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v1, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v11, v2, v4, v6, v7, v8, v9], [v1, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v1, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v1, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0010","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe581ba073ac703f5944ada35148485e2d79d388374daa0e7d126cb6dc5d3bcc","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156010,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [[v0, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [[v0, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 2, [[v0, v2], [v1, v3]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0011","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ada9c37c88a24515ab388fb7095bc904c39c716f0e0f4494e6aab3c0c06265","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156011,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [[v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [[v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0012","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"098e6d6aaeec67715207e59910b277532c194c3b7227d1b2d942f1292239b178","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156012,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [[v0, v2, v4], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0013","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc65fe070048f21c33888bd47c35764df093d9b3f8db8f5df4b0cec0aed53f76","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156013,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0014","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f152500b55141cd74eb71d7b522ca2725c41cf06d20a1a1b5f778484de0bdf31","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156014,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [[v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0015","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"753bcbc17c630fbe5968b00ce8d6db3c67b2238e13fdeaf6e2cdd2a4cd4a4984","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156015,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0016","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf5c24b47ea732e0e8ef266ecfb9dc3b5c77e43b744f9a69e6525f667509cf8","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156016,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v1, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v1, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v1, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v1, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v1, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v1, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v1, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v1, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v1, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v1, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v1, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v1, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v1, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v1, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v1, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v1, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v1, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v1, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v1, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v1, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v1, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v1, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v1, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v1, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v1, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v1, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v1, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v1, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v1, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v1, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v1, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v1, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v1, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0017","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6355870a02d6ae5f6919aa031a4eeeddb6d71a71af2a903f43339c43ef947309","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156017,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v1, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v1, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v1, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v1, v10, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v10, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v1, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v1, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v1, v10, v3, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v1, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v1, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v1, v10, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v1, v10, v3, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v1, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v1, v10, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v1, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v1, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v1, v10, v3, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v1, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v1, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v1, v10, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v1, v10, v3, v5, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v1, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v1, v10, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0018","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f12235a07e27df0fd9d69df67c8a6715f926491e3e98fbe83a45df64991b8d10","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156018,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v1, v10, v11, v3, v5, v6, v8], [v0, v1, v10, v11, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v11, v2, v3, v5, v6, v8], [v1, v10, v11, v2, v3, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v0, v10, v11, v2, v4, v5, v6, v8], [v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v1, v10, v11, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v5, v7, v8], [v1, v10, v11, v3, v5, v7, v8], [v0, v1, v10, v11, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v11, v2, v3, v5, v7, v8], [v1, v10, v11, v2, v3, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v0, v10, v11, v2, v4, v5, v7, v8], [v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v1, v10, v11, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v0, v10, v11, v2, v4, v6, v7, v8], [v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v1, v10, v11, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v1, v10, v11, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9], [v0, v1, v11, v3, v5, v7, v9], [v1, v10, v11, v3, v5, v7, v9], [v0, v1, v10, v11, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v11, v2, v3, v5, v7, v9], [v1, v11, v2, v3, v5, v7, v9], [v0, v1, v11, v2, v3, v5, v7, v9], [v0, v10, v11, v2, v3, v5, v7, v9], [v1, v10, v11, v2, v3, v5, v7, v9], [v0, v1, 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v9], [v0, v1, v11, v2, v4, v6, v8, v9], [v0, v10, v11, v2, v4, v6, v8, v9], [v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v1, v11, v3, v4, v6, v8, v9], [v0, v1, v11, v3, v4, v6, v8, v9], [v1, v10, v11, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v11, v2, v3, v4, v6, v8, v9], [v1, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v1, v11, v3, v5, v6, v8, v9], [v0, v1, v11, v3, v5, v6, v8, v9], [v1, v10, v11, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v11, v2, v3, v5, v6, v8, v9], [v1, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v11, v2, v4, v5, v6, v8, v9], [v1, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v1, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v1, v11, v3, v5, v7, v8, v9], [v0, v1, v11, v3, v5, v7, v8, v9], [v1, v10, v11, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v11, v2, v3, v5, v7, v8, v9], [v1, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v11, v2, v4, v5, v7, v8, v9], [v1, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v1, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v11, v2, v4, v6, v7, v8, v9], [v1, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v1, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v1, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [[v0, v10, v2, v4, v6, v8], [v0, v1, v10, v2, v4, v6, v8], [v0, v10, v11, v2, v4, v6, v8], [v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v11, v2, v4, v6, v8], [v0, v1, v10, v3, v4, v6, v8], [v1, v10, v11, v3, v4, v6, v8], [v0, v1, v10, v11, v3, v4, v6, v8], [v0, v10, v2, v3, v4, v6, v8], [v0, v1, v10, v2, v3, v4, v6, v8], [v0, v10, v11, v2, v3, v4, v6, v8], [v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v8], [v0, v1, v10, v3, v5, v6, v8], [v1, v10, v11, v3, v5, v6, v8], [v0, v1, v10, v11, v3, v5, v6, v8], [v0, v10, v2, v3, v5, v6, v8], [v0, v1, v10, v2, v3, v5, v6, v8], [v0, v10, v11, v2, v3, v5, v6, v8], [v1, v10, v11, v2, v3, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v8], [v0, v10, v2, v4, v5, v6, v8], [v0, v1, v10, v2, v4, v5, v6, v8], [v0, v10, v11, v2, v4, v5, v6, v8], [v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v8], [v0, v1, v10, v3, v4, v5, v6, v8], [v1, v10, v11, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v8], [v0, v10, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8], [v0, v1, v10, v3, v5, v7, v8], [v1, v10, v11, v3, v5, v7, v8], [v0, v1, v10, v11, v3, v5, v7, v8], [v0, v10, v2, v3, v5, v7, v8], [v0, v1, v10, v2, v3, v5, v7, v8], [v0, v10, v11, v2, v3, v5, v7, v8], [v1, v10, v11, v2, v3, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v7, v8], [v0, v10, v2, v4, v5, v7, v8], [v0, v1, v10, v2, v4, v5, v7, v8], [v0, v10, v11, v2, v4, v5, v7, v8], [v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v7, v8], [v0, v1, v10, v3, v4, v5, v7, v8], [v1, v10, v11, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v7, v8], [v0, v10, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8], [v0, v10, v2, v4, v6, v7, v8], [v0, v1, v10, v2, v4, v6, v7, v8], [v0, v10, v11, v2, v4, v6, v7, v8], [v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v6, v7, v8], [v0, v1, v10, v3, v4, v6, v7, v8], [v1, v10, v11, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v6, v7, v8], [v0, v10, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8], [v0, v1, v10, v3, v5, v6, v7, v8], [v1, v10, v11, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v5, v6, v7, v8], [v0, v10, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8], [v0, v10, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8], [v0, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v10, v3, v5, v7, v9], [v1, v11, v3, v5, v7, v9], [v0, v1, v11, v3, v5, v7, v9], [v1, v10, v11, v3, v5, v7, v9], [v0, v1, v10, v11, v3, v5, v7, v9], [v0, v10, v2, v3, v5, v7, v9], [v0, v1, v10, v2, v3, v5, v7, v9], [v0, v11, v2, v3, v5, v7, v9], [v1, v11, v2, v3, v5, v7, v9], [v0, v1, v11, v2, v3, v5, v7, v9], [v0, v10, v11, v2, v3, v5, v7, v9], [v1, v10, v11, v2, v3, v5, v7, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v9], [v0, v10, v2, v4, v5, v7, v9], [v0, v1, v10, v2, v4, v5, v7, v9], [v0, v11, v2, v4, v5, v7, v9], [v1, v11, v2, v4, v5, v7, v9], [v0, v1, v11, v2, v4, v5, v7, v9], [v0, v10, v11, v2, v4, v5, v7, v9], [v1, v10, v11, v2, v4, v5, v7, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v9], [v0, v1, v10, v3, v4, v5, v7, v9], [v1, v11, v3, v4, v5, v7, v9], [v0, v1, v11, v3, v4, v5, v7, v9], [v1, v10, v11, v3, v4, v5, v7, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v9], [v0, v10, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v9], [v0, v11, v2, v3, v4, v5, v7, v9], [v1, v11, v2, v3, v4, v5, v7, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v9], [v0, v10, v2, v4, v6, v7, v9], [v0, v1, v10, v2, v4, v6, v7, v9], [v0, v11, v2, v4, v6, v7, v9], [v1, v11, v2, v4, v6, v7, v9], [v0, v1, v11, v2, v4, v6, v7, v9], [v0, v10, v11, v2, v4, v6, v7, v9], [v1, v10, v11, v2, v4, v6, v7, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v9], [v0, v1, v10, v3, v4, v6, v7, v9], [v1, v11, v3, v4, v6, v7, v9], [v0, v1, v11, v3, v4, v6, v7, v9], [v1, v10, v11, v3, v4, v6, v7, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v9], [v0, v10, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v9], [v0, v11, v2, v3, v4, v6, v7, v9], [v1, v11, v2, v3, v4, v6, v7, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v9], [v0, v1, v10, v3, v5, v6, v7, v9], [v1, v11, v3, v5, v6, v7, v9], [v0, v1, v11, v3, v5, v6, v7, v9], [v1, v10, v11, v3, v5, v6, v7, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v9], [v0, v10, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v9], [v0, v11, v2, v3, v5, v6, v7, v9], [v1, v11, v2, v3, v5, v6, v7, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v9], [v0, v10, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v9], [v0, v11, v2, v4, v5, v6, v7, v9], [v1, v11, v2, v4, v5, v6, v7, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v9], [v1, v11, v3, v4, v5, v6, v7, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v11, v2, v4, v6, v8, v9], [v1, v11, v2, v4, v6, v8, v9], [v0, v1, v11, v2, v4, v6, v8, v9], [v0, v10, v11, v2, v4, v6, v8, v9], [v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v1, v11, v3, v4, v6, v8, v9], [v0, v1, v11, v3, v4, v6, v8, v9], [v1, v10, v11, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v11, v2, v3, v4, v6, v8, v9], [v1, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v1, v11, v3, v5, v6, v8, v9], [v0, v1, v11, v3, v5, v6, v8, v9], [v1, v10, v11, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v11, v2, v3, v5, v6, v8, v9], [v1, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v11, v2, v4, v5, v6, v8, v9], [v1, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v1, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v1, v11, v3, v5, v7, v8, v9], [v0, v1, v11, v3, v5, v7, v8, v9], [v1, v10, v11, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v11, v2, v3, v5, v7, v8, v9], [v1, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v11, v2, v4, v5, v7, v8, v9], [v1, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v1, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v11, v2, v4, v6, v7, v8, v9], [v1, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v1, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v1, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0019","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b35b5e4f770c3a119cf54a477768df8a9c8f1ddc5607b1f64049edb3610c1d","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156019,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0020","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1605443a9e356a4b0515699cbc4110b93e9a4f649ce9121c81b4c7e841531727","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156020,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0021","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b84d7e88ca73b5418099ab2cdf9804e42662f3be90cbb6119ed04109d0c4d6ce","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156021,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0022","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d66c61b0090838399f9a037fa6899a2436039294fc0c45922fa4fd7a2af58ac9","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156022,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0023","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dc7e10dd54f77b41a640bc810777900d0bf85f1b858b5647b9a198f2ed85c1d","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156023,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0024","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfb48fe163e97803c60cc36895c0739c3ea1ec6e67a5b7a60ae4a9e0d833f47e","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156024,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0025","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd6648af96c53570fe4e6fb6fc143ee6115e0af6e9d9004835dd00a4850f3e47","family_id":"qf-mod46-mincover-exhaustive-minimizers","seed":202611156025,"semantic_template":"qf-mod46-mincover-exhaustive-minimizers/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v0, v9], [v0, v1, v9], [v0, v2, v9], [v0, v1, v2, v9], [v0, v3, v9], [v0, v1, v3, v9], [v0, v2, v3, v9], [v0, v1, v2, v3, v9], [v0, v4, v9], [v0, v1, v4, v9], [v0, v2, v4, v9], [v0, v1, v2, v4, v9], [v0, v3, v4, v9], [v0, v1, v3, v4, v9], [v0, v2, v3, v4, v9], [v0, v1, v2, v3, v4, v9], [v0, v5, v9], [v0, v1, v5, v9], [v0, v2, v5, v9], [v0, v1, v2, v5, v9], [v0, v3, v5, v9], [v0, v1, v3, v5, v9], [v0, v2, v3, v5, v9], [v0, v1, v2, v3, v5, v9], [v0, v4, v5, v9], [v0, v1, v4, v5, v9], [v0, v2, v4, v5, v9], [v0, v1, v2, v4, v5, v9], [v0, v3, v4, v5, v9], [v0, v1, v3, v4, v5, v9], [v0, v2, v3, v4, v5, v9], [v0, v1, v2, v3, v4, v5, v9], [v0, v6, v9], [v0, v1, v6, v9], [v0, v2, v6, v9], [v0, v1, v2, v6, v9], [v0, v3, v6, v9], [v0, v1, v3, v6, v9], [v0, v2, v3, v6, v9], [v0, v1, v2, v3, v6, v9], [v0, v4, v6, v9], [v0, v1, v4, v6, v9], [v0, v2, v4, v6, v9], [v0, v1, v2, v4, v6, v9], [v0, v3, v4, v6, v9], [v0, v1, v3, v4, v6, v9], [v0, v2, v3, v4, v6, v9], [v0, v1, v2, v3, v4, v6, v9], [v0, v5, v6, v9], [v0, v1, v5, v6, v9], [v0, v2, v5, v6, v9], [v0, v1, v2, v5, v6, v9], [v0, v3, v5, v6, v9], [v0, v1, v3, v5, v6, v9], [v0, v2, v3, v5, v6, v9], [v0, v1, v2, v3, v5, v6, v9], [v0, v4, v5, v6, v9], [v0, v1, v4, v5, v6, v9], [v0, v2, v4, v5, v6, v9], [v0, v1, v2, v4, v5, v6, v9], [v0, v3, v4, v5, v6, v9], [v0, v1, v3, v4, v5, v6, v9], [v0, v2, v3, v4, v5, v6, v9], [v0, v1, v2, v3, v4, v5, v6, v9], [v0, v7, v9], [v0, v1, v7, v9], [v0, v2, v7, v9], [v0, v1, v2, v7, v9], [v0, v3, v7, v9], [v0, v1, v3, v7, v9], [v0, v2, v3, v7, v9], [v0, v1, v2, v3, v7, v9], [v0, v4, v7, v9], [v0, v1, v4, v7, v9], [v0, v2, v4, v7, v9], [v0, v1, v2, v4, v7, v9], [v0, v3, v4, v7, v9], [v0, v1, v3, v4, v7, v9], [v0, v2, v3, v4, v7, v9], [v0, v1, v2, v3, v4, v7, v9], [v0, v5, v7, v9], [v0, v1, v5, v7, v9], [v0, v2, v5, v7, v9], [v0, v1, v2, v5, v7, v9], [v0, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v4, v5, v7, v9], [v0, v1, v4, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v6, v7, v9], [v0, v1, v6, v7, v9], [v0, v2, v6, v7, v9], [v0, v1, v2, v6, v7, v9], [v0, v3, v6, v7, v9], [v0, v1, v3, v6, v7, v9], [v0, v2, v3, v6, v7, v9], [v0, v1, v2, v3, v6, v7, v9], [v0, v4, v6, v7, v9], [v0, v1, v4, v6, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v5, v6, v7, v9], [v0, v1, v5, v6, v7, v9], [v0, v2, v5, v6, v7, v9], [v0, v1, v2, v5, v6, v7, v9], [v0, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v4, v5, v6, v7, v9], [v0, v1, v4, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v8, v9], [v0, v1, v8, v9], [v0, v2, v8, v9], [v0, v1, v2, v8, v9], [v0, v3, v8, v9], [v0, v1, v3, v8, v9], [v0, v2, v3, v8, v9], [v0, v1, v2, v3, v8, v9], [v0, v4, v8, v9], [v0, v1, v4, v8, v9], [v0, v2, v4, v8, v9], [v0, v1, v2, v4, v8, v9], [v0, v3, v4, v8, v9], [v0, v1, v3, v4, v8, v9], [v0, v2, v3, v4, v8, v9], [v0, v1, v2, v3, v4, v8, v9], [v0, v5, v8, v9], [v0, v1, v5, v8, v9], [v0, v2, v5, v8, v9], [v0, v1, v2, v5, v8, v9], [v0, v3, v5, v8, v9], [v0, v1, v3, v5, v8, v9], [v0, v2, v3, v5, v8, v9], [v0, v1, v2, v3, v5, v8, v9], [v0, v4, v5, v8, v9], [v0, v1, v4, v5, v8, v9], [v0, v2, v4, v5, v8, v9], [v0, v1, v2, v4, v5, v8, v9], [v0, v3, v4, v5, v8, v9], [v0, v1, v3, v4, v5, v8, v9], [v0, v2, v3, v4, v5, v8, v9], [v0, v1, v2, v3, v4, v5, v8, v9], [v0, v6, v8, v9], [v0, v1, v6, v8, v9], [v0, v2, v6, v8, v9], [v0, v1, v2, v6, v8, v9], [v0, v3, v6, v8, v9], [v0, v1, v3, v6, v8, v9], [v0, v2, v3, v6, v8, v9], [v0, v1, v2, v3, v6, v8, v9], [v0, v4, v6, v8, v9], [v0, v1, v4, v6, v8, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v5, v6, v8, v9], [v0, v1, v5, v6, v8, v9], [v0, v2, v5, v6, v8, v9], [v0, v1, v2, v5, v6, v8, v9], [v0, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v4, v5, v6, v8, v9], [v0, v1, v4, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v7, v8, v9], [v0, v1, v7, v8, v9], [v0, v2, v7, v8, v9], [v0, v1, v2, v7, v8, v9], [v0, v3, v7, v8, v9], [v0, v1, v3, v7, v8, v9], [v0, v2, v3, v7, v8, v9], [v0, v1, v2, v3, v7, v8, v9], [v0, v4, v7, v8, v9], [v0, v1, v4, v7, v8, v9], [v0, v2, v4, v7, v8, v9], [v0, v1, v2, v4, v7, v8, v9], [v0, v3, v4, v7, v8, v9], [v0, v1, v3, v4, v7, v8, v9], [v0, v2, v3, v4, v7, v8, v9], [v0, v1, v2, v3, v4, v7, v8, v9], [v0, v5, v7, v8, v9], [v0, v1, v5, v7, v8, v9], [v0, v2, v5, v7, v8, v9], [v0, v1, v2, v5, v7, v8, v9], [v0, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v4, v5, v7, v8, v9], [v0, v1, v4, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v6, v7, v8, v9], [v0, v1, v6, v7, v8, v9], [v0, v2, v6, v7, v8, v9], [v0, v1, v2, v6, v7, v8, v9], [v0, v3, v6, v7, v8, v9], [v0, v1, v3, v6, v7, v8, v9], [v0, v2, v3, v6, v7, v8, v9], [v0, v1, v2, v3, v6, v7, v8, v9], [v0, v4, v6, v7, v8, v9], [v0, v1, v4, v6, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v5, v6, v7, v8, v9], [v0, v1, v5, v6, v7, v8, v9], [v0, v2, v5, v6, v7, v8, v9], [v0, v1, v2, v5, v6, v7, v8, v9], [v0, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v4, v5, v6, v7, v8, v9], [v0, v1, v4, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)","explanation":"Retain all covers in source subset order, then minimum cardinality and all covers attaining it. This complete bounded search is not a polynomial-time algorithm claim, approximation guarantee or use of Konig theorem. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [[v0], [v0, v1], [v0, v2], [v0, v1, v2], [v0, v3], [v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v4], [v0, v1, v4], [v0, v2, v4], [v0, v1, v2, v4], [v0, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v0, v1, v2, v3, v4], [v0, v5], [v0, v1, v5], [v0, v2, v5], [v0, v1, v2, v5], [v0, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v4, v5], [v0, v1, v4, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v0, v6], [v0, v1, v6], [v0, v2, v6], [v0, v1, v2, v6], [v0, v3, v6], [v0, v1, v3, v6], [v0, v2, v3, v6], [v0, v1, v2, v3, v6], [v0, v4, v6], [v0, v1, v4, v6], [v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v5, v6], [v0, v1, v5, v6], [v0, v2, v5, v6], [v0, v1, v2, v5, v6], [v0, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v4, v5, v6], [v0, v1, v4, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v0, v7], [v0, v1, v7], [v0, v2, v7], [v0, v1, v2, v7], [v0, v3, v7], [v0, v1, v3, v7], [v0, v2, v3, v7], [v0, v1, v2, v3, v7], [v0, v4, v7], [v0, v1, v4, v7], [v0, v2, v4, v7], [v0, v1, v2, v4, v7], [v0, v3, v4, v7], [v0, v1, v3, v4, v7], [v0, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v0, v5, v7], [v0, v1, v5, v7], [v0, v2, v5, v7], [v0, v1, v2, v5, v7], [v0, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v4, v5, v7], [v0, v1, v4, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v6, v7], [v0, v1, v6, v7], [v0, v2, v6, v7], [v0, v1, v2, v6, v7], [v0, v3, v6, v7], [v0, v1, v3, v6, v7], [v0, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v0, v4, v6, v7], [v0, v1, v4, v6, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v5, v6, v7], [v0, v1, v5, v6, v7], [v0, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v0, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7], [v0, v8], [v0, v1, v8], [v0, v2, v8], [v0, v1, v2, v8], [v0, v3, v8], [v0, v1, v3, v8], [v0, v2, v3, v8], [v0, v1, v2, v3, v8], [v0, v4, v8], [v0, v1, v4, v8], [v0, v2, v4, v8], [v0, v1, v2, v4, v8], [v0, v3, v4, v8], [v0, v1, v3, v4, v8], [v0, v2, v3, v4, v8], [v0, v1, v2, v3, v4, v8], [v0, v5, v8], [v0, v1, v5, v8], [v0, v2, v5, v8], [v0, v1, v2, v5, v8], [v0, v3, v5, v8], [v0, v1, v3, v5, v8], [v0, v2, v3, v5, v8], [v0, v1, v2, v3, v5, v8], [v0, v4, v5, v8], [v0, v1, v4, v5, v8], [v0, v2, v4, v5, v8], [v0, v1, v2, v4, v5, v8], [v0, v3, v4, v5, v8], [v0, v1, v3, v4, v5, v8], [v0, v2, v3, v4, v5, v8], [v0, v1, v2, v3, v4, v5, v8], [v0, v6, v8], [v0, v1, v6, v8], [v0, v2, v6, v8], [v0, v1, v2, v6, v8], [v0, v3, v6, v8], [v0, v1, v3, v6, v8], [v0, v2, v3, v6, v8], [v0, v1, v2, v3, v6, v8], [v0, v4, v6, v8], [v0, v1, v4, v6, v8], [v0, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v0, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v0, v5, v6, v8], [v0, v1, v5, v6, v8], [v0, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v0, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v0, v7, v8], [v0, v1, v7, v8], [v0, v2, v7, v8], [v0, v1, v2, v7, v8], [v0, v3, v7, v8], [v0, v1, v3, v7, v8], [v0, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v0, v4, v7, v8], [v0, v1, v4, v7, v8], [v0, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v0, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v0, v5, v7, v8], [v0, v1, v5, v7, v8], [v0, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v0, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v6, v7, v8], [v0, v1, v6, v7, v8], [v0, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v0, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v0, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v0, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8], [v0, v9], [v0, v1, v9], [v0, v2, v9], [v0, v1, v2, v9], [v0, v3, v9], [v0, v1, v3, v9], [v0, v2, v3, v9], [v0, v1, v2, v3, v9], [v0, v4, v9], [v0, v1, v4, v9], [v0, v2, v4, v9], [v0, v1, v2, v4, v9], [v0, v3, v4, v9], [v0, v1, v3, v4, v9], [v0, v2, v3, v4, v9], [v0, v1, v2, v3, v4, v9], [v0, v5, v9], [v0, v1, v5, v9], [v0, v2, v5, v9], [v0, v1, v2, v5, v9], [v0, v3, v5, v9], [v0, v1, v3, v5, v9], [v0, v2, v3, v5, v9], [v0, v1, v2, v3, v5, v9], [v0, v4, v5, v9], [v0, v1, v4, v5, v9], [v0, v2, v4, v5, v9], [v0, v1, v2, v4, v5, v9], [v0, v3, v4, v5, v9], [v0, v1, v3, v4, v5, v9], [v0, v2, v3, v4, v5, v9], [v0, v1, v2, v3, v4, v5, v9], [v0, v6, v9], [v0, v1, v6, v9], [v0, v2, v6, v9], [v0, v1, v2, v6, v9], [v0, v3, v6, v9], [v0, v1, v3, v6, v9], [v0, v2, v3, v6, v9], [v0, v1, v2, v3, v6, v9], [v0, v4, v6, v9], [v0, v1, v4, v6, v9], [v0, v2, v4, v6, v9], [v0, v1, v2, v4, v6, v9], [v0, v3, v4, v6, v9], [v0, v1, v3, v4, v6, v9], [v0, v2, v3, v4, v6, v9], [v0, v1, v2, v3, v4, v6, v9], [v0, v5, v6, v9], [v0, v1, v5, v6, v9], [v0, v2, v5, v6, v9], [v0, v1, v2, v5, v6, v9], [v0, v3, v5, v6, v9], [v0, v1, v3, v5, v6, v9], [v0, v2, v3, v5, v6, v9], [v0, v1, v2, v3, v5, v6, v9], [v0, v4, v5, v6, v9], [v0, v1, v4, v5, v6, v9], [v0, v2, v4, v5, v6, v9], [v0, v1, v2, v4, v5, v6, v9], [v0, v3, v4, v5, v6, v9], [v0, v1, v3, v4, v5, v6, v9], [v0, v2, v3, v4, v5, v6, v9], [v0, v1, v2, v3, v4, v5, v6, v9], [v0, v7, v9], [v0, v1, v7, v9], [v0, v2, v7, v9], [v0, v1, v2, v7, v9], [v0, v3, v7, v9], [v0, v1, v3, v7, v9], [v0, v2, v3, v7, v9], [v0, v1, v2, v3, v7, v9], [v0, v4, v7, v9], [v0, v1, v4, v7, v9], [v0, v2, v4, v7, v9], [v0, v1, v2, v4, v7, v9], [v0, v3, v4, v7, v9], [v0, v1, v3, v4, v7, v9], [v0, v2, v3, v4, v7, v9], [v0, v1, v2, v3, v4, v7, v9], [v0, v5, v7, v9], [v0, v1, v5, v7, v9], [v0, v2, v5, v7, v9], [v0, v1, v2, v5, v7, v9], [v0, v3, v5, v7, v9], [v0, v1, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v1, v2, v3, v5, v7, v9], [v0, v4, v5, v7, v9], [v0, v1, v4, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v1, v2, v4, v5, v7, v9], [v0, v3, v4, v5, v7, v9], [v0, v1, v3, v4, v5, v7, v9], [v0, v2, v3, v4, v5, v7, v9], [v0, v1, v2, v3, v4, v5, v7, v9], [v0, v6, v7, v9], [v0, v1, v6, v7, v9], [v0, v2, v6, v7, v9], [v0, v1, v2, v6, v7, v9], [v0, v3, v6, v7, v9], [v0, v1, v3, v6, v7, v9], [v0, v2, v3, v6, v7, v9], [v0, v1, v2, v3, v6, v7, v9], [v0, v4, v6, v7, v9], [v0, v1, v4, v6, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v1, v2, v4, v6, v7, v9], [v0, v3, v4, v6, v7, v9], [v0, v1, v3, v4, v6, v7, v9], [v0, v2, v3, v4, v6, v7, v9], [v0, v1, v2, v3, v4, v6, v7, v9], [v0, v5, v6, v7, v9], [v0, v1, v5, v6, v7, v9], [v0, v2, v5, v6, v7, v9], [v0, v1, v2, v5, v6, v7, v9], [v0, v3, v5, v6, v7, v9], [v0, v1, v3, v5, v6, v7, v9], [v0, v2, v3, v5, v6, v7, v9], [v0, v1, v2, v3, v5, v6, v7, v9], [v0, v4, v5, v6, v7, v9], [v0, v1, v4, v5, v6, v7, v9], [v0, v2, v4, v5, v6, v7, v9], [v0, v1, v2, v4, v5, v6, v7, v9], [v0, v3, v4, v5, v6, v7, v9], [v0, v1, v3, v4, v5, v6, v7, v9], [v0, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v9], [v0, v8, v9], [v0, v1, v8, v9], [v0, v2, v8, v9], [v0, v1, v2, v8, v9], [v0, v3, v8, v9], [v0, v1, v3, v8, v9], [v0, v2, v3, v8, v9], [v0, v1, v2, v3, v8, v9], [v0, v4, v8, v9], [v0, v1, v4, v8, v9], [v0, v2, v4, v8, v9], [v0, v1, v2, v4, v8, v9], [v0, v3, v4, v8, v9], [v0, v1, v3, v4, v8, v9], [v0, v2, v3, v4, v8, v9], [v0, v1, v2, v3, v4, v8, v9], [v0, v5, v8, v9], [v0, v1, v5, v8, v9], [v0, v2, v5, v8, v9], [v0, v1, v2, v5, v8, v9], [v0, v3, v5, v8, v9], [v0, v1, v3, v5, v8, v9], [v0, v2, v3, v5, v8, v9], [v0, v1, v2, v3, v5, v8, v9], [v0, v4, v5, v8, v9], [v0, v1, v4, v5, v8, v9], [v0, v2, v4, v5, v8, v9], [v0, v1, v2, v4, v5, v8, v9], [v0, v3, v4, v5, v8, v9], [v0, v1, v3, v4, v5, v8, v9], [v0, v2, v3, v4, v5, v8, v9], [v0, v1, v2, v3, v4, v5, v8, v9], [v0, v6, v8, v9], [v0, v1, v6, v8, v9], [v0, v2, v6, v8, v9], [v0, v1, v2, v6, v8, v9], [v0, v3, v6, v8, v9], [v0, v1, v3, v6, v8, v9], [v0, v2, v3, v6, v8, v9], [v0, v1, v2, v3, v6, v8, v9], [v0, v4, v6, v8, v9], [v0, v1, v4, v6, v8, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v3, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v5, v6, v8, v9], [v0, v1, v5, v6, v8, v9], [v0, v2, v5, v6, v8, v9], [v0, v1, v2, v5, v6, v8, v9], [v0, v3, v5, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v4, v5, v6, v8, v9], [v0, v1, v4, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v7, v8, v9], [v0, v1, v7, v8, v9], [v0, v2, v7, v8, v9], [v0, v1, v2, v7, v8, v9], [v0, v3, v7, v8, v9], [v0, v1, v3, v7, v8, v9], [v0, v2, v3, v7, v8, v9], [v0, v1, v2, v3, v7, v8, v9], [v0, v4, v7, v8, v9], [v0, v1, v4, v7, v8, v9], [v0, v2, v4, v7, v8, v9], [v0, v1, v2, v4, v7, v8, v9], [v0, v3, v4, v7, v8, v9], [v0, v1, v3, v4, v7, v8, v9], [v0, v2, v3, v4, v7, v8, v9], [v0, v1, v2, v3, v4, v7, v8, v9], [v0, v5, v7, v8, v9], [v0, v1, v5, v7, v8, v9], [v0, v2, v5, v7, v8, v9], [v0, v1, v2, v5, v7, v8, v9], [v0, v3, v5, v7, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v4, v5, v7, v8, v9], [v0, v1, v4, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v3, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v6, v7, v8, v9], [v0, v1, v6, v7, v8, v9], [v0, v2, v6, v7, v8, v9], [v0, v1, v2, v6, v7, v8, v9], [v0, v3, v6, v7, v8, v9], [v0, v1, v3, v6, v7, v8, v9], [v0, v2, v3, v6, v7, v8, v9], [v0, v1, v2, v3, v6, v7, v8, v9], [v0, v4, v6, v7, v8, v9], [v0, v1, v4, v6, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v5, v6, v7, v8, v9], [v0, v1, v5, v6, v7, v8, v9], [v0, v2, v5, v6, v7, v8, v9], [v0, v1, v2, v5, v6, v7, v8, v9], [v0, v3, v5, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v4, v5, v6, v7, v8, v9], [v0, v1, v4, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]], 1, [[v0]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)."},"skills":["Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0026","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9356db80c39d695015d8d114c44354acb3dd1a0d5b419771063470f20727c80","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156026,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3]], candidate cover=[v0, v1, v2, v3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0027","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf9f8b0d994509690d0b5a0e7d57f25a73d83528f1112ebc915b49b80265fbee","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156027,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]], candidate cover=[v0, v1, v2, v3, v4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], Yes, 5, 2, [v1, v3], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], Yes, 5, 2, [v1, v3], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0028","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f96418c5c4ef559549c5c922ea61bf70ca03ccf2cd97815e08e821edcc5b3d","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156028,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], candidate cover=[v0, v1, v2, v3, v4, v5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0029","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f229930a0e1e3b6d06446d9be30d2a13674b8d0e24fe78eb9ce90c28f432ce17","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156029,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], candidate cover=[v0, v1, v2, v3, v4, v5, v6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 3, [v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 3, [v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0030","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7601fa18060913506070b06f09f4f45e5c3d45f6f7735103b3beefe08372289d","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156030,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0031","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e261fba51c10009cdb02a48b6822312785170d51e1a98b314ee3b8f18a1e02","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156031,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 4, [v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 4, [v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0032","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b18d42c254ae6c2641a134d2a04479329a8b5b0f77bfd4240b1dd62af6120f2b","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156032,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 5, [v0, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 5, [v0, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0033","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308b9342f96367fc68f7a83a8e3278be2a1a73bc8155233c67bdcccbd4f58565","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156033,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 11, 5, [v1, v3, v5, v7, v9], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 11, 5, [v1, v3, v5, v7, v9], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0034","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c68ebeaff82797117c2bd794a7f2d09eb57a92b08deafaf569ebfc8c971aeb7","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156034,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 12, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 12, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0035","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3905c063f1f759940edd460cc85b469a07518d23558eea82eefb018007b0332","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156035,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]], candidate cover=[v0, v1, v2, v3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0036","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc801c82a68ab74d921e2145dbfdca305341d17958b8837a40bde333d55d8eef","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156036,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]], candidate cover=[v0, v1, v2, v3, v4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], Yes, 5, 3, [v0, v1, v3], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], Yes, 5, 3, [v0, v1, v3], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0037","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20f4b5c874afcf27fed6a968a42a0c61f5271071d820516841623ae8382a790","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156037,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]], candidate cover=[v0, v1, v2, v3, v4, v5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0038","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e9366fb588a71239b1ab0e9304841c0ce12d1d9592e2ad588e342775fea5876","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156038,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 4, [v0, v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 4, [v0, v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0039","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ff68ea80119a8b7f93f4b4f033d86520266322b2b49faadce28da1d2430c7b","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156039,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0040","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c32418c7fb28fa4a314ee760f2ee90e19a80a88a094e988e9cfab4c28d6afc06","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156040,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 5, [v0, v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 5, [v0, v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0041","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a117241ea28f07a28d55eee25ce5e41384656d57e6fc68685d42972aaee0b916","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156041,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 5, [v0, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 5, [v0, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0042","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cdf561a963d8fba82c14d014379f68c1e7963907327ae9ffd99cbe1a2428bd8","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156042,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 11, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 11, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0043","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32dfd5efa0251dc8048c13a216c6913faa7be8105f6832981f337ace19f4e947","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156043,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 12, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 12, 6, [v0, v10, v2, v4, v6, v8], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0044","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df78acad2bc94073e78fde94c76d869c4d498d85d79a7b1552cee2d7d24fcd44","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156044,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3]], candidate cover=[v0, v1, v2, v3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], Yes, 4, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3], Yes, 4, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0045","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0aa0b2995b5e48f37c90bb967ff177dd9f01143f96c71f72ec92df06b46e270","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156045,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]], candidate cover=[v0, v1, v2, v3, v4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], Yes, 5, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], Yes, 5, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0046","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0376677268264d1addfb1b2ec8a9a55c299720c1db9414ae9d880208dbffcfca","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156046,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], candidate cover=[v0, v1, v2, v3, v4, v5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], Yes, 6, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], Yes, 6, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0047","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93a27b8868b4dd1368cf86be9d94cc1cdc0dd83330ea06a6710f1491c3357dbd","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156047,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], candidate cover=[v0, v1, v2, v3, v4, v5, v6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0048","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ee33f387d4c72cfe35847e5b521da5548861711f5a170dca6150f907955c28","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156048,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0049","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d840f29793e7c57ecffb3ec92662e649a0ba83e535110d912bfcd5096c815ed","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156049,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0050","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d36e3ea3138244ea6504370a1c368d6321e8e049c47ab0b6600389e3967c3f8f","family_id":"qf-mod46-mincover-nonminimum-witness","seed":202611156050,"semantic_template":"qf-mod46-mincover-nonminimum-witness/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"An Explicit Smaller Cover Refutes Minimum Cardinality","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)","explanation":"Require the candidate to be a cover but strictly larger than the complete minimum. Retain its size and the first canonical minimum cover as a smaller witness. Coverage alone does not imply minimum or even inclusion-minimal status. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)."},"skills":["An Explicit Smaller Cover Refutes Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0051","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89ebf51b724d0609bbaa2b63e49d9b34a086860540cf1779c75c7138849fa9dc","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156051,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1]], candidate cover=[l0_0, l0_1]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0052","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7a673d2841f5254802ef87ee8794ef8aab8d44c79c3e321bc6346434e29e49f","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156052,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2]], candidate cover=[l0_0, l0_1, l0_2]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0053","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ece6911b30974946d66fa597e7a15d6a33088b3575a7e9959293bbd0d08a7124","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156053,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], candidate cover=[l0_0, l0_1, l0_2, l0_3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0054","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9f1eb461a41735bf3db8f6bc03267973bcbd199623ce068735fbd0f00aecf7b","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156054,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0055","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f864eaf1bc9fd2ddd583c605f29f1b013bbd690f89a71c1dcb49b119e17dce1c","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156055,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0056","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe3d8634b107f37b5bb532168681444d1d3ef1e439b37f09fab1cb89e6f50160","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156056,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0057","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d5393c4181d5f4521ecdf5dc04599bcf23af765eab9fafdadd8f0a93afa80ed","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156057,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0058","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60f22816d0c6bc2c7b72345f6780be68fe9528eb0ab899cbc3ef62ed803dfed7","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156058,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0059","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"852d804cf14c10118c0030b8aee6c5bc67b6dadbc075cfa130204acef002ffa8","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156059,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0060","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a93c7e3c76093f2ba7be03583159aa8b055ed9da4df6c5d6a231f3a125b7e84","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156060,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_10\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9], [c0, l0_10]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0061","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d47b71f0ec5ba54b4ad80c7ea76f705ed26a704cf99b849bebb3998f1acb9164","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156061,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]], candidate cover=[l0_0, l0_1, l1_0, l1_1]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0062","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a8307538c87b4d808f2171f17cfb74f750cb89574037f2f44d1507d653bf0f9","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156062,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0063","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"980a51954d9faae0b6b4930dfe35cd3eaea16c6245d471f0466347b30cb338c1","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156063,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0064","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5ddbba475b24b39edddddd47479054932e277288164385a96a82079524a1582","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156064,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0065","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40afcd6a2e597c3c9a12857bee87d548988bf49933f34814988f277cb5f5ff7e","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156065,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0066","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8fec2b81f4a1ca2bb032866cac66d3d66defbdb01cdf2d608407a084215f506","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156066,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0067","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"061c2e103b93b7e2c2a2d01e73bb9fb541799c6582b8b8d29d00b5b3218b8850","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156067,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6], [c1, l1_7]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0068","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd4c1c7288235012c8cf77cf2f67ac8abdd19cde7b8554d0268717ad5c09f8b","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156068,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0069","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3feadc1f3a8de7f74dc5405bd47f756b46ac29e699f053f727eedde6b5a66b8b","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156069,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0070","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"358a10af8bc0d2600215a53879fc7828196297c17b0c0a8ffc34e99e644632fd","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156070,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0071","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e724931f03d408fd3bbe294f82ab1c37286f44204ddf96dae1018d2b5ba2098","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156071,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0072","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02ec982d9a515ec30de1faf4b44f18f16b6ef0bb2769a308aaf92690b7e0d115","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156072,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0073","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04d10bdb23219c09d86f563b27ac775ac096622de29bfc858f363f3255a25346","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156073,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0074","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77414a668080923d6877da2105eb3f760951c468fb80db2976dcdaa67c57e321","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156074,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0075","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459864dd33231dead89df72e07534879ea749342869999960010afe10e61c6c7","family_id":"qf-mod46-mincover-minimal-not-minimum","seed":202611156075,"semantic_template":"qf-mod46-mincover-minimal-not-minimum/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover","stem":"Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)","explanation":"Every single-vertex deletion must lose coverage, while another complete cover is strictly smaller. Retain deletion rows, minimum size and a smaller witness. Inclusion-minimal, minimum and maximal are different notions. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)."},"skills":["A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0076","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5892f3bd936d51b13b7ff59881ca2c914c57aebd7c2bb7d7ad348deaa0cbc8f7","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156076,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0077","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f388da1f3265ce118fb5d3faeec5396fed1b1eb4a60d82c52cf4d93067e600b8","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156077,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0078","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eaad849cd4017ca108813ddb5de3d116adfcdfd4b40a09bd477067ee943326d","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156078,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0079","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff84bab0a35f52d1e1b20021537c039758d3aaad6a54300e6edc680318a3558e","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156079,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0080","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c65d93b08b570d721620509e8e3a48a7acb86fbfb4e099b602954bbc553f0ab0","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156080,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0081","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bae32cfd5818c80de833d1b94fcf446fd4bef8ae67d95d0afe942a4065be195","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156081,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0082","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0c62a98806c6556917af1d1dc5d32af5db575e6d389bf0bc4eeb5499c696122","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156082,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0083","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d35260f2dec8c4de03912bd61478bd760ea447bbdbb7a71b33235f78990a1f7","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156083,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0084","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccab99d121cea27c03fd36feb962661744c320a890d6f6d1a857d9317077ebb2","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156084,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0085","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c92b9ac9eb57993f9cc0ced5ba09ad8cd486996ef30e2e3ceb7b791d991873a7","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156085,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0086","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b858f90515f9d5c13bb5e5b833631dbaabfcd842e506ba7e097f9d3a241d3a6d","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156086,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0087","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e497ecaba0cb92f9df0451cef9302eca044624ed0407c69bf626b5aeab4f9957","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156087,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[],\"mode\":\"empty_graph\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0088","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fb7ba01cc73c04f475c06fe0b3e036a5e909c724bcf7869a90cd28e8082aa66","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156088,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v1\",\"v2\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v1, v2]], 2, [[v0, v1], [v0, v2], [v1, v2]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v2], [v1, v2]], 2, [[v0, v1], [v0, v2], [v1, v2]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0089","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bd10c9b1fc2398ea89c9f9c5315397e08075fa61c363ac9d3ccd2e31ebbaa0b","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156089,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], 2, [[v0, v2], [v1, v3]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], 2, [[v0, v2], [v1, v3]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0090","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f83d5d3963dde1eb708f8c440d15cde0960c9f0857087371f4d81412896f8901","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156090,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0091","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61a8120c281a8edb2835c73b1c5d4a098da5b986f423f784992405a0c8b8afbe","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156091,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], 3, [[v0, v2, v4], [v1, v3, v5]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0092","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b7da37baac7071819389407d00f1b0afdfd12594f6a47823e8d0d7af2e262bf","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156092,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0093","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d32d96f346370b5cd2dae817e9dc5b4e6a43d955dff314d7ce9f3fc3ad115e","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156093,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0094","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0455556fa874c39fffd3af6b66290387745313f03cf5c7fcab24e0a7b4f9c345","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156094,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0095","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b040ba654ad5b59e76bbcf96961acb9a9749ed874801b5b4484e3ec1efad2134","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156095,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 5, [[v0, v2, v4, v6, v8], [v1, v3, v5, v7, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0096","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbe727dbd94d4d99127f58a6f711a746160eb41e513b8e0dd28de7615740517d","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156096,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0097","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6560b485d6a7a029ce6b018dd58be28f14d408514c5eb2c723d112d57170d4cc","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156097,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0098","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28288adfc84a9dde0b9500a5ff6d8977331b74c86bfc86da8040e2348c442769","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156098,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v2\",\"v3\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v1, v2], [v1, v3], [v2, v3]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v1, v2], [v1, v3], [v2, v3]], 3, [[v0, v1, v2], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v1, v2], [v1, v3], [v2, v3]], 3, [[v0, v1, v2], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0099","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00c03639e070dfd572c99a4ee881fcd657f5a2bafec41f0aa8e5226b87b17ddb","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156099,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v3\",\"v4\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r06-minimum-vertex-cover-0100","batch_id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"948d7e84e09f719320d016c6f7cd9f99cacb652a6b0dee255320f90d87eb495c","family_id":"qf-mod46-mincover-empty-multiple-boundary","seed":202611156100,"semantic_template":"qf-mod46-mincover-empty-multiple-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v1\",\"v4\"],[\"v1\",\"v5\"],[\"v2\",\"v3\"],[\"v2\",\"v4\"],[\"v2\",\"v5\"],[\"v3\",\"v4\"],[\"v3\",\"v5\"],[\"v4\",\"v5\"]],\"mode\":\"multiple_minima\",\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several","stem":"Use complete unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v1, v2], [v1, v3], [v1, v4], [v1, v5], [v2, v3], [v2, v4], [v2, v5], [v3, v4], [v3, v5], [v4, v5]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).","answer":{"kind":"expression","value":"(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v1, v2], [v1, v3], [v1, v4], [v1, v5], [v2, v3], [v2, v4], [v2, v5], [v3, v4], [v3, v5], [v4, v5]], 5, [[v0, v1, v2, v3, v4], [v0, v1, v2, v3, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)","explanation":"For empty_graph require no edges, not an empty vertex carrier, and retain the unique empty minimum cover. For multiple_minima retain every minimum solution and verify at least two. No weighted/directed extension or Konig theorem is asserted. The exact result, retaining the requested order, is (multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v1, v2], [v1, v3], [v1, v4], [v1, v5], [v2, v3], [v2, v4], [v2, v5], [v3, v4], [v3, v5], [v4, v5]], 5, [[v0, v1, v2, v3, v4], [v0, v1, v2, v3, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)."},"skills":["Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}