第 26070 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-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3"]}。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-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3"]}。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-minimal-not-minimum]

derived = [[candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate]]

source roles = [[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]]

derived = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]]

source roles = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]], 512]

derived = [[3, 7, 11, 15, 19, 23, 27, 30, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 94, 95, 99, 103, 107, 111, 115, 119, 123, 126, 127, 131, 135, 139, 143, 147, 151, 155, 158, 159, 163, 167, 171, 175, 179, 183, 187, 190, 191, 195, 199, 203, 207, 211, 215, 219, 222, 223, 227, 231, 235, 239, 243, 247, 251, 254, 255, 259, 263, 267, 271, 275, 279, 283, 286, 287, 291, 295, 299, 303, 307, 311, 315, 318, 319, 323, 327, 331, 335, 339, 343, 347, 350, 351, 355, 359, 363, 367, 371, 375, 379, 382, 383, 387, 391, 395, 399, 403, 407, 411, 414, 415, 419, 423, 427, 431, 435, 439, 443, 446, 447, 451, 455, 459, 463, 467, 471, 475, 478, 479, 481, 483, 485, 487, 489, 491, 493, 495, 497, 499, 501, 503, 505, 507, 508, 509, 510, 511], [[0, 0], [1, 3], [2, 0], [4, 1], [5, 3], [6, 1], [8, 0], [9, 3], [10, 0], [12, 2], [13, 3], [14, 2], [16, 0], [17, 3], [18, 0], [20, 1], [21, 3], [22, 1], [24, 0], [25, 3], [26, 0], [28, 3], [29, 3], [32, 0], [33, 4], [34, 0], [36, 1], [37, 4], [38, 1], [40, 0], [41, 4], [42, 0], [44, 2], [45, 4], [46, 2], [48, 0], [49, 4], [50, 0], [52, 1], [53, 4], [54, 1], [56, 0], [57, 4], [58, 0], [60, 4], [61, 4], [64, 0], [65, 3], [66, 0], [68, 1], [69, 3], [70, 1], [72, 0], [73, 3], [74, 0], [76, 2], [77, 3], [78, 2], [80, 0], [81, 3], [82, 0], [84, 1], [85, 3], [86, 1], [88, 0], [89, 3], [90, 0], [92, 3], [93, 3], [96, 0], [97, 5], [98, 0], [100, 1], [101, 5], [102, 1], [104, 0], [105, 5], [106, 0], [108, 2], [109, 5], [110, 2], [112, 0], [113, 5], [114, 0], [116, 1], [117, 5], [118, 1], [120, 0], [121, 5], [122, 0], [124, 5], [125, 5], [128, 0], [129, 3], [130, 0], [132, 1], [133, 3], [134, 1], [136, 0], [137, 3], [138, 0], [140, 2], [141, 3], [142, 2], [144, 0], [145, 3], [146, 0], [148, 1], [149, 3], [150, 1], [152, 0], [153, 3], [154, 0], [156, 3], [157, 3], [160, 0], [161, 4], [162, 0], [164, 1], [165, 4], [166, 1], [168, 0], [169, 4], [170, 0], [172, 2], [173, 4], [174, 2], [176, 0], [177, 4], [178, 0], [180, 1], [181, 4], [182, 1], [184, 0], [185, 4], [186, 0], [188, 4], [189, 4], [192, 0], [193, 3], [194, 0], [196, 1], [197, 3], [198, 1], [200, 0], [201, 3], [202, 0], [204, 2], [205, 3], [206, 2], [208, 0], [209, 3], [210, 0], [212, 1], [213, 3], [214, 1], [216, 0], [217, 3], [218, 0], [220, 3], [221, 3], [224, 0], [225, 6], [226, 0], [228, 1], [229, 6], [230, 1], [232, 0], [233, 6], [234, 0], [236, 2], [237, 6], [238, 2], [240, 0], [241, 6], [242, 0], [244, 1], [245, 6], [246, 1], [248, 0], [249, 6], [250, 0], [252, 6], [253, 6], [256, 0], [257, 3], [258, 0], [260, 1], [261, 3], [262, 1], [264, 0], [265, 3], [266, 0], [268, 2], [269, 3], [270, 2], [272, 0], [273, 3], [274, 0], [276, 1], [277, 3], [278, 1], [280, 0], [281, 3], [282, 0], [284, 3], [285, 3], [288, 0], [289, 4], [290, 0], [292, 1], [293, 4], [294, 1], [296, 0], [297, 4], [298, 0], [300, 2], [301, 4], [302, 2], [304, 0], [305, 4], [306, 0], [308, 1], [309, 4], [310, 1], [312, 0], [313, 4], [314, 0], [316, 4], [317, 4], [320, 0], [321, 3], [322, 0], [324, 1], [325, 3], [326, 1], [328, 0], [329, 3], [330, 0], [332, 2], [333, 3], [334, 2], [336, 0], [337, 3], [338, 0], [340, 1], [341, 3], [342, 1], [344, 0], [345, 3], [346, 0], [348, 3], [349, 3], [352, 0], [353, 5], [354, 0], [356, 1], [357, 5], [358, 1], [360, 0], [361, 5], [362, 0], [364, 2], [365, 5], [366, 2], [368, 0], [369, 5], [370, 0], [372, 1], [373, 5], [374, 1], [376, 0], [377, 5], [378, 0], [380, 5], [381, 5], [384, 0], [385, 3], [386, 0], [388, 1], [389, 3], [390, 1], [392, 0], [393, 3], [394, 0], [396, 2], [397, 3], [398, 2], [400, 0], [401, 3], [402, 0], [404, 1], [405, 3], [406, 1], [408, 0], [409, 3], [410, 0], [412, 3], [413, 3], [416, 0], [417, 4], [418, 0], [420, 1], [421, 4], [422, 1], [424, 0], [425, 4], [426, 0], [428, 2], [429, 4], [430, 2], [432, 0], [433, 4], [434, 0], [436, 1], [437, 4], [438, 1], [440, 0], [441, 4], [442, 0], [444, 4], [445, 4], [448, 0], [449, 3], [450, 0], [452, 1], [453, 3], [454, 1], [456, 0], [457, 3], [458, 0], [460, 2], [461, 3], [462, 2], [464, 0], [465, 3], [466, 0], [468, 1], [469, 3], [470, 1], [472, 0], [473, 3], [474, 0], [476, 3], [477, 3], [480, 0], [482, 0], [484, 1], [486, 1], [488, 0], [490, 0], [492, 2], [494, 2], [496, 0], [498, 0], [500, 1], [502, 1], [504, 0], [506, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l0_2], [c0, c1, l0_0, l0_2], [c0, c1, l0_1, l0_2], [c1, l0_0, l0_1, l0_2], [c0, c1, l0_0, l0_1, l0_2], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l0_2, l1_0], [c0, c1, l0_0, l0_2, l1_0], [c0, c1, l0_1, l0_2, l1_0], [c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l0_2, l1_1], [c0, c1, l0_0, l0_2, l1_1], [c0, c1, l0_1, l0_2, l1_1], [c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_2, l1_0, l1_1], [c0, c1, l0_1, l0_2, l1_0, l1_1], [c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l1_2], [c0, c1, l0_0, l1_2], [c0, c1, l0_1, l1_2], [c0, c1, l0_0, l0_1, l1_2], [c0, c1, l0_2, l1_2], [c0, c1, l0_0, l0_2, l1_2], [c0, c1, l0_1, l0_2, l1_2], [c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l1_0, l1_2], [c0, c1, l0_0, l1_0, l1_2], [c0, c1, l0_1, l1_0, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_2], [c0, c1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2], [c0, c1, l1_1, l1_2], [c0, c1, l0_0, l1_1, l1_2], [c0, c1, l0_1, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_1, l1_2], [c0, c1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, c1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l1_0, l1_1, l1_2], [c0, c1, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l1_3], [c0, c1, l0_0, l1_3], [c0, c1, l0_1, l1_3], [c0, c1, l0_0, l0_1, l1_3], [c0, c1, l0_2, l1_3], [c0, c1, l0_0, l0_2, l1_3], [c0, c1, l0_1, l0_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_3], [c0, c1, l1_0, l1_3], [c0, c1, l0_0, l1_0, l1_3], [c0, c1, l0_1, l1_0, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_3], [c0, c1, l0_2, l1_0, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_3], [c0, c1, l1_1, l1_3], [c0, c1, l0_0, l1_1, l1_3], [c0, c1, l0_1, l1_1, l1_3], [c0, c1, l0_0, l0_1, l1_1, l1_3], [c0, c1, l0_2, l1_1, l1_3], [c0, c1, l0_0, l0_2, l1_1, l1_3], [c0, c1, l0_1, l0_2, l1_1, l1_3], [c1, l0_0, l0_1, l0_2, l1_1, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_3], [c0, c1, l1_0, l1_1, l1_3], [c0, c1, l0_0, l1_0, l1_1, l1_3], [c0, c1, l0_1, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3], [c0, c1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l1_2, l1_3], [c0, c1, l0_0, l1_2, l1_3], [c0, c1, l0_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_2, l1_3], [c0, c1, l0_2, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_2, l1_3], [c0, c1, l1_0, l1_2, l1_3], [c0, c1, l0_0, l1_0, l1_2, l1_3], [c0, c1, l0_1, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3], [c0, c1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l1_1, l1_2, l1_3], [c0, c1, l0_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3], [c0, c1, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_1, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], [c0, l1_0, l1_1, l1_2, l1_3], [c0, c1, l1_0, l1_1, l1_2, l1_3], [c0, l0_0, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3], [c0, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]], [0, 0, 1, 7, 22, 40, 44, 29, 9, 1], 2, [[c0, c1]]]

source roles = [[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]]]

derived = [[[[c0, l0_0], 是], [[c0, l0_1], 是], [[c0, l0_2], 是], [[c1, l1_0], 是], [[c1, l1_1], 是], [[c1, l1_2], 是], [[c1, l1_3], 是]], 7, 2, [c0, c1]]

source roles = [[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]]]

derived = [是, 2, [c0, c1], 否]

逐步反向核對

• 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

([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)

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。