Question 26090: An Explicit Smaller Cover Refutes Minimum Cardinality

Original question and givens

Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).

• G001 — Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-minimum-vertex-cover

Step-by-step derivation

source roles = [mincover-nonminimum-witness]

derived = [[candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate]]

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], [v8, v0]], [v0, v1, v2, v3, v4, v5, v6, v7, v8]]

derived = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v5, v6, v7, v8]]

source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], 512]

derived = [[171, 173, 175, 181, 183, 187, 189, 191, 213, 215, 219, 221, 223, 235, 237, 239, 245, 247, 251, 253, 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, 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], [85, 8], [86, 1], [87, 8], [88, 0], [89, 2], [90, 1], [91, 8], [92, 0], [93, 8], [94, 1], [95, 8], [96, 0], [97, 2], [98, 1], [99, 3], [100, 0], [101, 4], [102, 1], [103, 4], [104, 0], [105, 2], [106, 1], [107, 8], [108, 0], [109, 8], [110, 1], [111, 8], [112, 0], [113, 2], [114, 1], [115, 3], [116, 0], [117, 8], [118, 1], [119, 8], [120, 0], [121, 2], [122, 1], [123, 8], [124, 0], [125, 8], [126, 1], [127, 8], [128, 0], [129, 2], [130, 1], [131, 3], [132, 0], [133, 4], [134, 1], [135, 4], [136, 0], [137, 2], [138, 1], [139, 5], [140, 0], [141, 5], [142, 1], [143, 5], [144, 0], [145, 2], [146, 1], [147, 3], [148, 0], [149, 6], [150, 1], [151, 6], [152, 0], [153, 2], [154, 1], [155, 6], [156, 0], [157, 6], [158, 1], [159, 6], [160, 0], [161, 2], [162, 1], [163, 3], [164, 0], [165, 4], [166, 1], [167, 4], [168, 0], [169, 2], [170, 1], [172, 0], [174, 1], [176, 0], [177, 2], [178, 1], [179, 3], [180, 0], [182, 1], [184, 0], [185, 2], [186, 1], [188, 0], [190, 1], [192, 0], [193, 2], [194, 1], [195, 3], [196, 0], [197, 4], [198, 1], [199, 4], [200, 0], [201, 2], [202, 1], [203, 5], [204, 0], [205, 5], [206, 1], [207, 5], [208, 0], [209, 2], [210, 1], [211, 3], [212, 0], [214, 1], [216, 0], [217, 2], [218, 1], [220, 0], [222, 1], [224, 0], [225, 2], [226, 1], [227, 3], [228, 0], [229, 4], [230, 1], [231, 4], [232, 0], [233, 2], [234, 1], [236, 0], [238, 1], [240, 0], [241, 2], [242, 1], [243, 3], [244, 0], [246, 1], [248, 0], [249, 2], [250, 1], [252, 0], [254, 1], [256, 0], [257, 2], [258, 3], [259, 3], [260, 0], [261, 4], [262, 4], [263, 4], [264, 0], [265, 2], [266, 5], [267, 5], [268, 0], [269, 5], [270, 5], [271, 5], [272, 0], [273, 2], [274, 3], [275, 3], [276, 0], [277, 6], [278, 6], [279, 6], [280, 0], [281, 2], [282, 6], [283, 6], [284, 0], [285, 6], [286, 6], [287, 6], [288, 0], [289, 2], [290, 3], [291, 3], [292, 0], [293, 4], [294, 4], [295, 4], [296, 0], [297, 2], [298, 7], [299, 7], [300, 0], [301, 7], [302, 7], [303, 7], [304, 0], [305, 2], [306, 3], [307, 3], [308, 0], [309, 7], [310, 7], [311, 7], [312, 0], [313, 2], [314, 7], [315, 7], [316, 0], [317, 7], [318, 7], [319, 7], [320, 0], [321, 2], [322, 3], [323, 3], [324, 0], [325, 4], [326, 4], [327, 4], [328, 0], [329, 2], [330, 5], [331, 5], [332, 0], [333, 5], [334, 5], [335, 5], [336, 0], [337, 2], [338, 3], [339, 3], [340, 0], [344, 0], [345, 2], [348, 0], [352, 0], [353, 2], [354, 3], [355, 3], [356, 0], [357, 4], [358, 4], [359, 4], [360, 0], [361, 2], [364, 0], [368, 0], [369, 2], [370, 3], [371, 3], [372, 0], [376, 0], [377, 2], [380, 0], [384, 0], [385, 2], [386, 3], [387, 3], [388, 0], [389, 4], [390, 4], [391, 4], [392, 0], [393, 2], [394, 5], [395, 5], [396, 0], [397, 5], [398, 5], [399, 5], [400, 0], [401, 2], [402, 3], [403, 3], [404, 0], [405, 6], [406, 6], [407, 6], [408, 0], [409, 2], [410, 6], [411, 6], [412, 0], [413, 6], [414, 6], [415, 6], [416, 0], [417, 2], [418, 3], [419, 3], [420, 0], [421, 4], [422, 4], [423, 4], [424, 0], [425, 2], [428, 0], [432, 0], [433, 2], [434, 3], [435, 3], [436, 0], [440, 0], [441, 2], [444, 0], [448, 0], [449, 2], [450, 3], [451, 3], [452, 0], [453, 4], [454, 4], [455, 4], [456, 0], [457, 2], [458, 5], [459, 5], [460, 0], [461, 5], [462, 5], [463, 5], [464, 0], [465, 2], [466, 3], [467, 3], [468, 0], [472, 0], [473, 2], [476, 0], [480, 0], [481, 2], [482, 3], [483, 3], [484, 0], [485, 4], [486, 4], [487, 4], [488, 0], [489, 2], [492, 0], [496, 0], [497, 2], [498, 3], [499, 3], [500, 0], [504, 0], [505, 2], [508, 0]], [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v0, v1, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v0, 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, 0, 9, 30, 27, 9, 1], 5, [[v0, v1, v3, v5, v7], [v0, v2, v3, v5, v7], [v0, v2, v4, v5, v7], [v0, v2, v4, v6, v7], [v0, v2, v4, v6, v8], [v1, v2, v4, v6, v8], [v1, v3, v4, v6, v8], [v1, v3, v5, v6, v8], [v1, v3, v5, v7, v8]]]

source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]]]

derived = [[[[v0, v1], Yes], [[v0, v8], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], 9, 5, [v0, v1, v3, v5, v7]]

Reverse checks

• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, 9, 5, [v0, v1, v3, v5, v7], No, explicit_smaller_cover_refutes_minimum_status)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.