Question 26088: An Explicit Smaller Cover Refutes Minimum Cardinality

Original question and givens

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

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

source roles = [[v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], 128]

derived = [[43, 45, 47, 53, 55, 59, 61, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127], [[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], [44, 0], [46, 1], [48, 0], [49, 2], [50, 1], [51, 3], [52, 0], [54, 1], [56, 0], [57, 2], [58, 1], [60, 0], [62, 1], [64, 0], [65, 2], [66, 3], [67, 3], [68, 0], [69, 4], [70, 4], [71, 4], [72, 0], [73, 2], [74, 5], [75, 5], [76, 0], [77, 5], [78, 5], [79, 5], [80, 0], [81, 2], [82, 3], [83, 3], [84, 0], [88, 0], [89, 2], [92, 0], [96, 0], [97, 2], [98, 3], [99, 3], [100, 0], [101, 4], [102, 4], [103, 4], [104, 0], [105, 2], [108, 0], [112, 0], [113, 2], [114, 3], [115, 3], [116, 0], [120, 0], [121, 2], [124, 0]], [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v0, v1, v2, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [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]], [0, 0, 0, 0, 7, 14, 7, 1], 4, [[v0, v1, v3, v5], [v0, v2, v3, v5], [v0, v2, v4, v5], [v0, v2, v4, v6], [v1, v2, v4, v6], [v1, v3, v4, v6], [v1, v3, v5, v6]]]

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

derived = [[[[v0, v1], Yes], [[v0, v6], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], 7, 4, [v0, v1, v3, v5]]

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], Yes, 7, 4, [v0, v1, v3, v5], 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.