第 26007 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":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[]}。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":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[]}。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, v6], [], null]

derived = [[v0, v1, v2, v3, v4, v5, v6], [], null]

source roles = [[v0, v1, v2, v3, v4, v5, v6], [], 128]

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127], [], [[], [v0], [v1], [v0, v1], [v2], [v0, v2], [v1, v2], [v0, v1, v2], [v3], [v0, v3], [v1, v3], [v0, v1, v3], [v2, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v4], [v0, v4], [v1, v4], [v0, v1, v4], [v2, v4], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v3, v4], [v0, v3, v4], [v1, v3, v4], [v0, v1, v3, v4], [v2, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v5], [v0, v5], [v1, v5], [v0, v1, v5], [v2, v5], [v0, v2, v5], [v1, v2, v5], [v0, v1, v2, v5], [v3, v5], [v0, v3, v5], [v1, v3, v5], [v0, v1, v3, v5], [v2, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v4, v5], [v0, v4, v5], [v1, v4, v5], [v0, v1, v4, v5], [v2, v4, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v3, v4, v5], [v0, v3, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v2, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v6], [v0, v6], [v1, v6], [v0, v1, v6], [v2, v6], [v0, v2, v6], [v1, v2, v6], [v0, v1, v2, v6], [v3, v6], [v0, v3, v6], [v1, v3, v6], [v0, v1, v3, v6], [v2, v3, v6], [v0, v2, v3, v6], [v1, v2, v3, v6], [v0, v1, v2, v3, v6], [v4, v6], [v0, v4, v6], [v1, v4, v6], [v0, v1, v4, v6], [v2, v4, v6], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v3, v4, v6], [v0, v3, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v2, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v5, v6], [v0, v5, v6], [v1, v5, v6], [v0, v1, v5, v6], [v2, v5, v6], [v0, v2, v5, v6], [v1, v2, v5, v6], [v0, v1, v2, v5, v6], [v3, v5, v6], [v0, v3, v5, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v2, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v4, v5, v6], [v0, v4, v5, v6], [v1, v4, v5, v6], [v0, v1, v4, v5, v6], [v2, v4, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v3, v4, v5, v6], [v0, v3, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v2, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6]], [1, 7, 21, 35, 35, 21, 7, 1], 0, [[]]]

source roles = [empty_graph, [v0, v1, v2, v3, v4, v5, v6], []]

derived = [0, [[]], 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

(empty_graph, [], 0, [[]], 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。