Question 25408: Every Undirected Edge Needs at Least One Selected Endpoint

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], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v8, v9, v10]. 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. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, 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], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v8, v9, v10]. 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. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, 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

Step-by-step derivation

source roles = [vcover-complete-edge-audit]

derived = [[vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate]]

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], [v9, v10]], [v0, v1, v2, v3, v4, v5, v6, v8, v9, v10]]

derived = [[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, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

source roles = [0, [v0, v1], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [1, [v1, v2], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [2, [v10, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [3, [v2, v3], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [4, [v3, v4], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [5, [v4, v5], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [6, [v5, v6], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [7, [v6, v7], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, No], Yes]

source roles = [8, [v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[No, Yes], Yes]

source roles = [9, [v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9]]

derived = [[Yes, Yes], Yes]

source roles = [[[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]]]

derived = [Yes, null, []]

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

([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, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

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.