Question 26029: Minimum Vertex Covers Are Selected Only after Complete Subset Coverage Audits

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]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, 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]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (vertices, edges, all_covers, minimum_size, minimum_covers, 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-exhaustive-minimizers]

derived = [[vertices, edges, all_covers, minimum_size, minimum_covers, certificate]]

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

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

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

derived = [[42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 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, 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], [44, 0], [48, 0], [49, 1], [50, 2], [51, 2], [52, 0], [56, 0], [57, 1], [60, 0], [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]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, 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, 1, 10, 15, 7, 1], 3, [[v1, v3, v5]]]

Reverse checks

• 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], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [[v1, v3, v5], [v0, v1, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, 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]], 3, [[v1, v3, v5]], all_vertex_subsets_are_audited_before_selecting_all_minimizers)

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.