# 第 26081 題：最小頂點覆蓋 [返回所在分頁 131](https://ocme.evemisslab.com/zh-Hant/questions/page/131/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations/#q-20261015-advance-20261115-r06-minimum-vertex-cover-0031) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0051-26081 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0031 ## 原題與已知條件 bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；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"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"]}。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-nonminimum-witness；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"]],"candidate":["v0","v1","v2","v3","v4","v5","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-minimum-vertex-cover](https://ocme.evemisslab.com/zh-Hant/mko/mko-minimum-vertex-cover/) ## 逐步推導 S001 · implies操作前 source roles = [mincover-nonminimum-witness] 操作後 derived = [[candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate]] 條件：無 本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。 步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001 S002 · implies操作前 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]], [v0, v1, v2, v3, v4, v5, v6, v7, v8]] 操作後 derived = [[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]], [v0, v1, v2, v3, v4, v5, v6, v7, v8]] 條件：無 非空carrier至多12；端點UTF16排序後edge族English排序，loop/反向duplicate拒絕。未提供candidate時保持null，不擅補空candidate。 步驟識別與依賴規則：Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · 依賴：G001, S001 S003 · implies操作前 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]], 512] 操作後 derived = [[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, 341, 342, 343, 346, 347, 349, 350, 351, 362, 363, 365, 366, 367, 373, 374, 375, 378, 379, 381, 382, 383, 426, 427, 429, 430, 431, 437, 438, 439, 442, 443, 445, 446, 447, 469, 470, 471, 474, 475, 477, 478, 479, 490, 491, 493, 494, 495, 501, 502, 503, 506, 507, 509, 510, 511], [[0, 0], [1, 1], [2, 2], [3, 2], [4, 0], [5, 3], [6, 3], [7, 3], [8, 0], [9, 1], [10, 4], [11, 4], [12, 0], [13, 4], [14, 4], [15, 4], [16, 0], [17, 1], [18, 2], [19, 2], [20, 0], [21, 5], [22, 5], [23, 5], [24, 0], [25, 1], [26, 5], [27, 5], [28, 0], [29, 5], [30, 5], [31, 5], [32, 0], [33, 1], [34, 2], [35, 2], [36, 0], [37, 3], [38, 3], [39, 3], [40, 0], [41, 1], [42, 6], [43, 6], [44, 0], [45, 6], [46, 6], [47, 6], [48, 0], [49, 1], [50, 2], [51, 2], [52, 0], [53, 6], [54, 6], [55, 6], [56, 0], [57, 1], [58, 6], [59, 6], [60, 0], [61, 6], [62, 6], [63, 6], [64, 0], [65, 1], [66, 2], [67, 2], [68, 0], [69, 3], [70, 3], [71, 3], [72, 0], [73, 1], [74, 4], [75, 4], [76, 0], [77, 4], [78, 4], [79, 4], [80, 0], [81, 1], [82, 2], [83, 2], [84, 0], [85, 7], [86, 7], [87, 7], [88, 0], [89, 1], [90, 7], [91, 7], [92, 0], [93, 7], [94, 7], [95, 7], [96, 0], [97, 1], [98, 2], [99, 2], [100, 0], [101, 3], [102, 3], [103, 3], [104, 0], [105, 1], [106, 7], [107, 7], [108, 0], [109, 7], [110, 7], [111, 7], [112, 0], [113, 1], [114, 2], [115, 2], [116, 0], [117, 7], [118, 7], [119, 7], [120, 0], [121, 1], [122, 7], [123, 7], [124, 0], [125, 7], [126, 7], [127, 7], [128, 0], [129, 1], [130, 2], [131, 2], [132, 0], [133, 3], [134, 3], [135, 3], [136, 0], [137, 1], [138, 4], [139, 4], [140, 0], [141, 4], [142, 4], [143, 4], [144, 0], [145, 1], [146, 2], [147, 2], [148, 0], [149, 5], [150, 5], [151, 5], [152, 0], [153, 1], [154, 5], [155, 5], [156, 0], [157, 5], [158, 5], [159, 5], [160, 0], [161, 1], [162, 2], [163, 2], [164, 0], [165, 3], [166, 3], [167, 3], [168, 0], [169, 1], [172, 0], [176, 0], [177, 1], [178, 2], [179, 2], [180, 0], [184, 0], [185, 1], [188, 0], [192, 0], [193, 1], [194, 2], [195, 2], [196, 0], [197, 3], [198, 3], [199, 3], [200, 0], [201, 1], [202, 4], [203, 4], [204, 0], [205, 4], [206, 4], [207, 4], [208, 0], [209, 1], [210, 2], [211, 2], [212, 0], [216, 0], [217, 1], [220, 0], [224, 0], [225, 1], [226, 2], [227, 2], [228, 0], [229, 3], [230, 3], [231, 3], [232, 0], [233, 1], [236, 0], [240, 0], [241, 1], [242, 2], [243, 2], [244, 0], [248, 0], [249, 1], [252, 0], [256, 0], [257, 1], [258, 2], [259, 2], [260, 0], [261, 3], [262, 3], [263, 3], [264, 0], [265, 1], [266, 4], [267, 4], [268, 0], [269, 4], [270, 4], [271, 4], [272, 0], [273, 1], [274, 2], [275, 2], [276, 0], [277, 5], [278, 5], [279, 5], [280, 0], [281, 1], [282, 5], [283, 5], [284, 0], [285, 5], [286, 5], [287, 5], [288, 0], [289, 1], [290, 2], [291, 2], [292, 0], [293, 3], [294, 3], [295, 3], [296, 0], [297, 1], [298, 6], [299, 6], [300, 0], [301, 6], [302, 6], [303, 6], [304, 0], [305, 1], [306, 2], [307, 2], [308, 0], [309, 6], [310, 6], [311, 6], [312, 0], [313, 1], [314, 6], [315, 6], [316, 0], [317, 6], [318, 6], [319, 6], [320, 0], [321, 1], [322, 2], [323, 2], [324, 0], [325, 3], [326, 3], [327, 3], [328, 0], [329, 1], [330, 4], [331, 4], [332, 0], [333, 4], [334, 4], [335, 4], [336, 0], [337, 1], [338, 2], [339, 2], [340, 0], [344, 0], [345, 1], [348, 0], [352, 0], [353, 1], [354, 2], [355, 2], [356, 0], [357, 3], [358, 3], [359, 3], [360, 0], [361, 1], [364, 0], [368, 0], [369, 1], [370, 2], [371, 2], [372, 0], [376, 0], [377, 1], [380, 0], [384, 0], [385, 1], [386, 2], [387, 2], [388, 0], [389, 3], [390, 3], [391, 3], [392, 0], [393, 1], [394, 4], [395, 4], [396, 0], [397, 4], [398, 4], [399, 4], [400, 0], [401, 1], [402, 2], [403, 2], [404, 0], [405, 5], [406, 5], [407, 5], [408, 0], [409, 1], [410, 5], [411, 5], [412, 0], [413, 5], [414, 5], [415, 5], [416, 0], [417, 1], [418, 2], [419, 2], [420, 0], [421, 3], [422, 3], [423, 3], [424, 0], [425, 1], [428, 0], [432, 0], [433, 1], [434, 2], [435, 2], [436, 0], [440, 0], [441, 1], [444, 0], [448, 0], [449, 1], [450, 2], [451, 2], [452, 0], [453, 3], [454, 3], [455, 3], [456, 0], [457, 1], [458, 4], [459, 4], [460, 0], [461, 4], [462, 4], [463, 4], [464, 0], [465, 1], [466, 2], [467, 2], [468, 0], [472, 0], [473, 1], [476, 0], [480, 0], [481, 1], [482, 2], [483, 2], [484, 0], [485, 3], [486, 3], [487, 3], [488, 0], [489, 1], [492, 0], [496, 0], [497, 1], [498, 2], [499, 2], [500, 0], [504, 0], [505, 1], [508, 0]], [[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], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v0, v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v0, v1, v3, v4, v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], [0, 0, 0, 0, 1, 15, 35, 28, 9, 1], 4, [[v1, v3, v5, v7]]] 條件：無 逐全部mask核所有edges；每個失敗mask保首uncovered edge index反例，通過者全列source順序。較小size計數0給下界，實際最小解給上界；全部ties依numeric mask保留，不改English/大小序。 步驟識別與依賴規則：Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · 依賴：G001, S002 S004 · implies操作前 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]]] 操作後 derived = [[[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]], 9, 4, [v1, v3, v5, v7]] 條件：無 先核candidate真cover；較小解取完整source清單的首minimum，不用任意小集合代替。 步驟識別與依賴規則：Mincover_actual_candidate_complete_cover_and_smaller_witness · 依賴：G001, S003 ## 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 ([v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, 9, 4, [v1, v3, v5, v7], 否, explicit_smaller_cover_refutes_minimum_status) 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。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c5e261fba51c10009cdb02a48b6822312785170d51e1a98b314ee3b8f18a1e02 - candidate record SHA-256: fa21f8f71148cacbb42f527e1f96036c8741f34ad915f7261b01020707b27b5d - semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3 - question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r06-minimum-vertex-cover-0031.json)