Question 26064: A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

Original question and givens

Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).

• G001 — Use complete unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, 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-minimal-not-minimum]

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

source roles = [[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]]

derived = [[c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]]

source roles = [[c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], 512]

derived = [[3, 7, 11, 14, 15, 19, 23, 27, 30, 31, 35, 39, 43, 46, 47, 51, 55, 59, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 94, 95, 99, 103, 107, 110, 111, 115, 119, 123, 126, 127, 131, 135, 139, 142, 143, 147, 151, 155, 158, 159, 163, 167, 171, 174, 175, 179, 183, 187, 190, 191, 195, 199, 203, 206, 207, 211, 215, 219, 222, 223, 227, 231, 235, 238, 239, 243, 247, 251, 254, 255, 259, 263, 267, 270, 271, 275, 279, 283, 286, 287, 291, 295, 299, 302, 303, 307, 311, 315, 318, 319, 323, 327, 331, 334, 335, 339, 343, 347, 350, 351, 355, 359, 363, 366, 367, 371, 375, 379, 382, 383, 387, 391, 395, 398, 399, 403, 407, 411, 414, 415, 419, 423, 427, 430, 431, 435, 439, 443, 446, 447, 451, 455, 459, 462, 463, 467, 471, 475, 478, 479, 483, 487, 491, 494, 495, 497, 499, 501, 503, 505, 507, 508, 509, 510, 511], [[0, 0], [1, 2], [2, 0], [4, 1], [5, 2], [6, 1], [8, 0], [9, 2], [10, 0], [12, 2], [13, 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, 2], [34, 0], [36, 1], [37, 2], [38, 1], [40, 0], [41, 2], [42, 0], [44, 2], [45, 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, 2], [66, 0], [68, 1], [69, 2], [70, 1], [72, 0], [73, 2], [74, 0], [76, 2], [77, 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, 2], [98, 0], [100, 1], [101, 2], [102, 1], [104, 0], [105, 2], [106, 0], [108, 2], [109, 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, 2], [130, 0], [132, 1], [133, 2], [134, 1], [136, 0], [137, 2], [138, 0], [140, 2], [141, 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, 2], [162, 0], [164, 1], [165, 2], [166, 1], [168, 0], [169, 2], [170, 0], [172, 2], [173, 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, 2], [194, 0], [196, 1], [197, 2], [198, 1], [200, 0], [201, 2], [202, 0], [204, 2], [205, 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, 2], [226, 0], [228, 1], [229, 2], [230, 1], [232, 0], [233, 2], [234, 0], [236, 2], [237, 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, 2], [258, 0], [260, 1], [261, 2], [262, 1], [264, 0], [265, 2], [266, 0], [268, 2], [269, 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, 2], [290, 0], [292, 1], [293, 2], [294, 1], [296, 0], [297, 2], [298, 0], [300, 2], [301, 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, 2], [322, 0], [324, 1], [325, 2], [326, 1], [328, 0], [329, 2], [330, 0], [332, 2], [333, 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, 2], [354, 0], [356, 1], [357, 2], [358, 1], [360, 0], [361, 2], [362, 0], [364, 2], [365, 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, 2], [386, 0], [388, 1], [389, 2], [390, 1], [392, 0], [393, 2], [394, 0], [396, 2], [397, 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, 2], [418, 0], [420, 1], [421, 2], [422, 1], [424, 0], [425, 2], [426, 0], [428, 2], [429, 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, 2], [450, 0], [452, 1], [453, 2], [454, 1], [456, 0], [457, 2], [458, 0], [460, 2], [461, 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], [481, 2], [482, 0], [484, 1], [485, 2], [486, 1], [488, 0], [489, 2], [490, 0], [492, 2], [493, 2], [496, 0], [498, 0], [500, 1], [502, 1], [504, 0], [506, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c1, l0_0, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c1, l0_0, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c1, l0_0, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l1_2], [c0, c1, l0_0, l1_2], [c0, c1, l0_1, l1_2], [c1, l0_0, l0_1, l1_2], [c0, c1, l0_0, l0_1, l1_2], [c0, c1, l1_0, l1_2], [c0, c1, l0_0, l1_0, l1_2], [c0, c1, l0_1, l1_0, l1_2], [c1, l0_0, l0_1, l1_0, l1_2], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_1, l1_2], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l1_3], [c0, c1, l0_0, l1_3], [c0, c1, l0_1, l1_3], [c1, l0_0, l0_1, l1_3], [c0, c1, l0_0, l0_1, l1_3], [c0, c1, l1_0, l1_3], [c0, c1, l0_0, l1_0, l1_3], [c0, c1, l0_1, l1_0, l1_3], [c1, l0_0, l0_1, l1_0, l1_3], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_1, l1_3], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_1, 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], [c1, l0_0, l0_1, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3], [c0, c1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3], [c1, 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, c1, l1_4], [c0, c1, l0_0, l1_4], [c0, c1, l0_1, l1_4], [c1, l0_0, l0_1, l1_4], [c0, c1, l0_0, l0_1, l1_4], [c0, c1, l1_0, l1_4], [c0, c1, l0_0, l1_0, l1_4], [c0, c1, l0_1, l1_0, l1_4], [c1, l0_0, l0_1, l1_0, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_4], [c0, c1, l1_1, l1_4], [c0, c1, l0_0, l1_1, l1_4], [c0, c1, l0_1, l1_1, l1_4], [c1, l0_0, l0_1, l1_1, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_4], [c0, c1, l1_0, l1_1, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_4], [c1, l0_0, l0_1, l1_0, l1_1, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_4], [c0, c1, l1_2, l1_4], [c0, c1, l0_0, l1_2, l1_4], [c0, c1, l0_1, l1_2, l1_4], [c1, l0_0, l0_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_2, l1_4], [c0, c1, l1_0, l1_2, l1_4], [c0, c1, l0_0, l1_0, l1_2, l1_4], [c0, c1, l0_1, l1_0, l1_2, l1_4], [c1, l0_0, l0_1, l1_0, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_4], [c0, c1, l1_1, l1_2, l1_4], [c0, c1, l0_0, l1_1, l1_2, l1_4], [c0, c1, l0_1, l1_1, l1_2, l1_4], [c1, l0_0, l0_1, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_4], [c0, c1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_4], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l1_3, l1_4], [c0, c1, l0_0, l1_3, l1_4], [c0, c1, l0_1, l1_3, l1_4], [c1, l0_0, l0_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_3, l1_4], [c0, c1, l1_0, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_3, l1_4], [c1, l0_0, l0_1, l1_0, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_4], [c0, c1, l1_1, l1_3, l1_4], [c0, c1, l0_0, l1_1, l1_3, l1_4], [c0, c1, l0_1, l1_1, l1_3, l1_4], [c1, l0_0, l0_1, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_4], [c0, c1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_4], [c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_4], [c0, c1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], [c0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]], [0, 0, 1, 8, 26, 45, 46, 29, 9, 1], 2, [[c0, c1]]]

source roles = [[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]]]

derived = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes], [[c1, l1_3], Yes], [[c1, l1_4], Yes]], 7, 2, [c0, c1]]

source roles = [[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]]]

derived = [Yes, 2, [c0, c1], No]

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)

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.