Question 26080: An Explicit Smaller Cover Refutes Minimum Cardinality

Original question and givens

Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7]. 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] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], candidate cover=[v0, v1, v2, v3, v4, v5, v6, v7]. 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], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v5, v6, v7]]

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

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

derived = [[85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 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, 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], [88, 0], [89, 1], [92, 0], [96, 0], [97, 1], [98, 2], [99, 2], [100, 0], [101, 3], [102, 3], [103, 3], [104, 0], [105, 1], [108, 0], [112, 0], [113, 1], [114, 2], [115, 2], [116, 0], [120, 0], [121, 1], [124, 0], [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]], [[v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, 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, 5, 20, 21, 8, 1], 4, [[v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6], [v1, v3, v5, v7]]]

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

derived = [[[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], 8, 4, [v0, v2, v4, v6]]

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], Yes, 8, 4, [v0, v2, v4, v6], 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.