第 26018 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","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]]}。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","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]]}。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, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]], null]

derived = [[v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], null]

source roles = [[v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 256]

derived = [[85, 87, 91, 93, 95, 107, 109, 111, 117, 119, 123, 125, 127, 170, 171, 173, 174, 175, 181, 182, 183, 186, 187, 189, 190, 191, 213, 214, 215, 218, 219, 221, 222, 223, 234, 235, 237, 238, 239, 245, 246, 247, 250, 251, 253, 254, 255], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [5, 4], [6, 1], [7, 4], [8, 0], [9, 2], [10, 1], [11, 5], [12, 0], [13, 5], [14, 1], [15, 5], [16, 0], [17, 2], [18, 1], [19, 3], [20, 0], [21, 6], [22, 1], [23, 6], [24, 0], [25, 2], [26, 1], [27, 6], [28, 0], [29, 6], [30, 1], [31, 6], [32, 0], [33, 2], [34, 1], [35, 3], [36, 0], [37, 4], [38, 1], [39, 4], [40, 0], [41, 2], [42, 1], [43, 7], [44, 0], [45, 7], [46, 1], [47, 7], [48, 0], [49, 2], [50, 1], [51, 3], [52, 0], [53, 7], [54, 1], [55, 7], [56, 0], [57, 2], [58, 1], [59, 7], [60, 0], [61, 7], [62, 1], [63, 7], [64, 0], [65, 2], [66, 1], [67, 3], [68, 0], [69, 4], [70, 1], [71, 4], [72, 0], [73, 2], [74, 1], [75, 5], [76, 0], [77, 5], [78, 1], [79, 5], [80, 0], [81, 2], [82, 1], [83, 3], [84, 0], [86, 1], [88, 0], [89, 2], [90, 1], [92, 0], [94, 1], [96, 0], [97, 2], [98, 1], [99, 3], [100, 0], [101, 4], [102, 1], [103, 4], [104, 0], [105, 2], [106, 1], [108, 0], [110, 1], [112, 0], [113, 2], [114, 1], [115, 3], [116, 0], [118, 1], [120, 0], [121, 2], [122, 1], [124, 0], [126, 1], [128, 0], [129, 2], [130, 3], [131, 3], [132, 0], [133, 4], [134, 4], [135, 4], [136, 0], [137, 2], [138, 5], [139, 5], [140, 0], [141, 5], [142, 5], [143, 5], [144, 0], [145, 2], [146, 3], [147, 3], [148, 0], [149, 6], [150, 6], [151, 6], [152, 0], [153, 2], [154, 6], [155, 6], [156, 0], [157, 6], [158, 6], [159, 6], [160, 0], [161, 2], [162, 3], [163, 3], [164, 0], [165, 4], [166, 4], [167, 4], [168, 0], [169, 2], [172, 0], [176, 0], [177, 2], [178, 3], [179, 3], [180, 0], [184, 0], [185, 2], [188, 0], [192, 0], [193, 2], [194, 3], [195, 3], [196, 0], [197, 4], [198, 4], [199, 4], [200, 0], [201, 2], [202, 5], [203, 5], [204, 0], [205, 5], [206, 5], [207, 5], [208, 0], [209, 2], [210, 3], [211, 3], [212, 0], [216, 0], [217, 2], [220, 0], [224, 0], [225, 2], [226, 3], [227, 3], [228, 0], [229, 4], [230, 4], [231, 4], [232, 0], [233, 2], [236, 0], [240, 0], [241, 2], [242, 3], [243, 3], [244, 0], [248, 0], [249, 2], [252, 0]], [[v0, v2, v4, v6], [v0, v1, v2, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], [0, 0, 0, 0, 2, 16, 20, 8, 1], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]]]

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

derived = [4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], 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, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], 4, [[v0, v2, v4, v6], [v1, v3, v5, v7]], 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。