第 26025 題：最小頂點覆蓋

原題與已知條件

bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5"],"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"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4","v5"],"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"]]}。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-minimum-vertex-cover

逐步推導

source roles = [mincover-empty-multiple-boundary]

derived = [[mode, edges, minimum_size, minimum_covers, mode_certificate, boundary]]

source roles = [[v0, v1, v2, v3, v4, v5], [[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]], null]

derived = [[v0, v1, v2, v3, v4, v5], [[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]], null]

source roles = [[v0, v1, v2, v3, v4, v5], [[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]], 64]

derived = [[31, 47, 55, 59, 61, 62, 63], [[0, 0], [1, 5], [2, 1], [3, 9], [4, 0], [5, 6], [6, 2], [7, 12], [8, 0], [9, 5], [10, 1], [11, 10], [12, 0], [13, 7], [14, 3], [15, 14], [16, 0], [17, 5], [18, 1], [19, 9], [20, 0], [21, 6], [22, 2], [23, 13], [24, 0], [25, 5], [26, 1], [27, 11], [28, 0], [29, 8], [30, 4], [32, 0], [33, 5], [34, 1], [35, 9], [36, 0], [37, 6], [38, 2], [39, 12], [40, 0], [41, 5], [42, 1], [43, 10], [44, 0], [45, 7], [46, 3], [48, 0], [49, 5], [50, 1], [51, 9], [52, 0], [53, 6], [54, 2], [56, 0], [57, 5], [58, 1], [60, 0]], [[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], [v0, v1, v2, v3, v4, v5]], [0, 0, 0, 0, 0, 6, 1], 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]]]

source roles = [multiple_minima, [v0, v1, v2, v3, v4, v5], [[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]]]

derived = [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]], 是]

逐步反向核對

• 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

(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]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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。