第 25415 題：頂點覆蓋

原題與已知條件

有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-vertex-cover

逐步推導

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], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[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, v2, v3, v4, v6, v7, v8]]

source roles = [0, [v0, v1], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 是], 是]

source roles = [1, [v0, v8], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 是], 是]

source roles = [2, [v1, v2], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 是], 是]

source roles = [3, [v2, v3], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 是], 是]

source roles = [4, [v3, v4], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 是], 是]

source roles = [5, [v4, v5], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[是, 否], 是]

source roles = [6, [v5, v6], [v0, v1, v2, v3, v4, v6, v7, v8]]

derived = [[否, 是], 是]

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

derived = [[是, 是], 是]

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

derived = [[是, 是], 是]

source roles = [[[[v0, v1], 是], [[v0, v8], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]]]

derived = [是, null, []]

逐步反向核對

• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

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

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。