# Question 25434: Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks [Return to catalogue page 128](https://ocme.evemisslab.com/en/questions/page/128/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r07-vertex-cover-foundations/#q-20261015-advance-20261114-r07-vertex-cover-0059) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0049-25434 · q-20261015-advance-20261114-r07-vertex-cover-0059 ## Original question and givens Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate). - G001 — Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-vertex-cover](https://ocme.evemisslab.com/en/mko/mko-vertex-cover/) ## Step-by-step derivation S001 · impliesBefore source roles = [vcover-empty-isolated-superset] After derived = [[edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate]] Conditions: none List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[v0, v1, v10, 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, v1, v2, v3, v4, v5, v6, v7, v8, v9]] Conditions: none The carrier is nonempty with at most12 vertices.Normalize endpoints by JS UTF16<,then source English canonical edge order.Reject loops/reversed duplicate edges;this is not a directed graph or edge cover. Step identifiers and dependenciesRule: Vcover_complete_simple_undirected_graph_UTF16_then_English_order · Uses: G001, S001 S003 · impliesBefore source roles = [0, [v0, v1], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S002 S004 · impliesBefore source roles = [1, [v1, v2], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S003 S005 · impliesBefore source roles = [2, [v2, v3], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S004 S006 · impliesBefore source roles = [3, [v3, v4], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S005 S007 · impliesBefore source roles = [4, [v4, v5], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S006 S008 · impliesBefore source roles = [5, [v5, v6], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S007 S009 · impliesBefore source roles = [6, [v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S008 S010 · impliesBefore source roles = [7, [v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S009 S011 · impliesBefore source roles = [8, [v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [[Yes, Yes], Yes] Conditions: none This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count. Step identifiers and dependenciesRule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S010 S012 · impliesBefore source roles = [[[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]]] After derived = [Yes, null, [v10]] Conditions: none Conjoin all edges and retain the first uncovered edge in source order,or null,notFalse.Isolated vertices have no incident coverage obligation and are optional. Step identifiers and dependenciesRule: Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · Uses: G001, S011 S013 · impliesBefore source roles = [[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]] After derived = [No, null, Yes] Conditions: none An edgeless graph permits the empty cover;otherwise this conditional field is null.If C covers and C is a subset of S,each selected endpoint remains in S,so S also covers.Equal supersets are allowed. Step identifiers and dependenciesRule: Vcover_empty_graph_conditional_NULL_and_cover_superset_monotonicity · Uses: G001, S012 ## Reverse checks - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3bb2c6f1654536bfddf3982a755b347b944df85426c8492cbf6076dc82cdcbdb - candidate record SHA-256: 21bbccef5c080664cf2242ab9265bc6227b1547810c3344325debf2b84b247a1 - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r07-vertex-cover-0059.json)