OCMEBilingual mathematics & derivations繁體中文

All Minimum Covers and Explicit Nonminimum Witnesses

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batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

Field-order and source-binding data

1. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

2. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

3. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

4. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

5. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

6. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

7. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

8. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

9. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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, 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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)

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, v7, v9], [v0, v1, v10, v11, 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, 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], [v1, v10, v3, v5, 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], [v1, 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], [v1, 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], [v1, v10, v3, 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], [v1, 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], [v1, 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], [v1, v10, v3, 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], [v1, 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], [v1, v10, v3, v5, 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], [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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

10. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

11. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

12. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

13. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

14. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

15. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

16. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

17. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

18. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

19. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

20. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

21. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

22. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

23. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

24. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

25. Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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).

View answer and explanation

([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], 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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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-exhaustive-minimizers · candidate

26. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

27. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4], Yes, 5, 2, [v1, v3], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

28. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

29. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 3, [v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

30. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

31. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 4, [v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

32. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

33. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

34. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

35. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3], Yes, 4, 2, [v0, v2], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

36. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4], Yes, 5, 3, [v0, v1, v3], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

37. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5], Yes, 6, 3, [v0, v2, v4], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

38. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 4, [v0, v1, v3, v5], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

39. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 4, [v0, v2, v4, v6], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

40. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 5, [v0, v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

41. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

42. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

43. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

44. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3], Yes, 4, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

45. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4], Yes, 5, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

46. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5], Yes, 6, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

47. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6], Yes, 7, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

48. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7], Yes, 8, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

49. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

50. An Explicit Smaller Cover Refutes Minimum Cardinality

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).

View answer and explanation

([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, 10, 1, [v0], No, explicit_smaller_cover_refutes_minimum_status)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-nonminimum-witness · candidate

51. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

52. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

53. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

54. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

55. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

56. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

57. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

58. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

59. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

60. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

61. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

62. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

63. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

64. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

65. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

66. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

67. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

68. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

69. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

70. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

71. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

72. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

73. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

74. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

75. A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

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).

View answer and explanation

([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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-minimal-not-minimum · candidate

76. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

77. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

78. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

79. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

80. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

81. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

82. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

83. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

84. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

85. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

86. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

87. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

88. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

89. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

90. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

91. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

92. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

93. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

94. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

95. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

96. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

97. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

98. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

99. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

100. Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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).

empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness.

View answer and explanation

(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)

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).

Reviewed step-by-step derivation

Minimum vertex cover · qf-mod46-mincover-empty-multiple-boundary · candidate

Canonical question JSONL · English JSONL · Validation report