第 25444 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-empty-isolated-superset；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"]}。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-empty-isolated-superset；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"]}。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-empty-isolated-superset]

derived = [[edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate]]

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

derived = [[v0, v1, v10, 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, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

source roles = [0, [v0, v1], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [1, [v0, v2], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [2, [v0, v3], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [3, [v0, v4], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [4, [v0, v5], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [5, [v0, v6], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

source roles = [6, [v0, v7], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]]

derived = [[是, 是], 是]

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

derived = [[是, 是], 是]

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

derived = [[是, 是], 是]

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

derived = [是, null, [v10]]

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

derived = [否, null, 是]

逐步反向核對

• S013 · 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。
• 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

(否, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

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。