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Question 26054: A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0051-26054 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0054

Original question and givens

Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_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, l0_2, l0_3, l0_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], candidate cover=[l0_0, l0_1, l0_2, l0_3, l0_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.

Step-by-step derivation

S001 · implies

Before

source roles = [mincover-minimal-not-minimum]

After

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

Conditions: none

List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[c0, l0_0, l0_1, l0_2, l0_3, l0_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], [l0_0, l0_1, l0_2, l0_3, l0_4]]

After

derived = [[c0, l0_0, l0_1, l0_2, l0_3, l0_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], [l0_0, l0_1, l0_2, l0_3, l0_4]]

Conditions: none

The carrier is nonempty with at most12 vertices;sort endpoints by UTF16 then edge families in English.Reject loops/reversed duplicates.An omitted candidate stays null,not an invented empty candidate.

Step identifiers and dependencies

Rule: Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · Uses: G001, S001

S003 · implies

Before

source roles = [[c0, l0_0, l0_1, l0_2, l0_3, l0_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]], 64]

After

derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63], [[0, 0], [2, 1], [4, 0], [6, 2], [8, 0], [10, 1], [12, 0], [14, 3], [16, 0], [18, 1], [20, 0], [22, 2], [24, 0], [26, 1], [28, 0], [30, 4], [32, 0], [34, 1], [36, 0], [38, 2], [40, 0], [42, 1], [44, 0], [46, 3], [48, 0], [50, 1], [52, 0], [54, 2], [56, 0], [58, 1], [60, 0]], [[c0], [c0, l0_0], [c0, l0_1], [c0, l0_0, l0_1], [c0, l0_2], [c0, l0_0, l0_2], [c0, l0_1, l0_2], [c0, l0_0, l0_1, l0_2], [c0, l0_3], [c0, l0_0, l0_3], [c0, l0_1, l0_3], [c0, l0_0, l0_1, l0_3], [c0, l0_2, l0_3], [c0, l0_0, l0_2, l0_3], [c0, l0_1, l0_2, l0_3], [c0, l0_0, l0_1, l0_2, l0_3], [c0, l0_4], [c0, l0_0, l0_4], [c0, l0_1, l0_4], [c0, l0_0, l0_1, l0_4], [c0, l0_2, l0_4], [c0, l0_0, l0_2, l0_4], [c0, l0_1, l0_2, l0_4], [c0, l0_0, l0_1, l0_2, l0_4], [c0, l0_3, l0_4], [c0, l0_0, l0_3, l0_4], [c0, l0_1, l0_3, l0_4], [c0, l0_0, l0_1, l0_3, l0_4], [c0, l0_2, l0_3, l0_4], [c0, l0_0, l0_2, l0_3, l0_4], [c0, l0_1, l0_2, l0_3, l0_4], [l0_0, l0_1, l0_2, l0_3, l0_4], [c0, l0_0, l0_1, l0_2, l0_3, l0_4]], [0, 1, 5, 10, 10, 6, 1], 1, [[c0]]]

Conditions: none

Audit every mask against every edge,retaining the first uncovered-edge refutation for each failure and every accepted set in source order.Zero smaller-size counts give a lower bound and actual minimum covers attain it.Keep all ties in numeric-mask order,not English/size order.

Step identifiers and dependencies

Rule: Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · Uses: G001, S002

S004 · implies

Before

source roles = [[l0_0, l0_1, l0_2, l0_3, l0_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]]]

After

derived = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes], [[c0, l0_4], Yes]], 5, 1, [c0]]

Conditions: none

First qualify an actual candidate cover.The smaller witness is the first minimum from the complete source list,not an arbitrary small set.

Step identifiers and dependencies

Rule: Mincover_actual_candidate_complete_cover_and_smaller_witness · Uses: G001, S003

S005 · implies

Before

source roles = [[l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]]]

After

derived = [Yes, 1, [c0], No]

Conditions: none

Every single deletion loses coverage,so the candidate is inclusion-minimal.But exhaustive search supplies a strictly smaller cover,so it is not minimum.Superset monotonicity makes deletion tests cover all proper-subset obligations,not maximum or optimal size.

Step identifiers and dependencies

Rule: Mincover_ALL_single_deletions_minimal_BUT_smaller_cover_exists · Uses: G001, S004

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, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, [c0], 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: b9f1eb461a41735bf3db8f6bc03267973bcbd199623ce068735fbd0f00aecf7b
  • candidate record SHA-256: a5ec2c48f0364693acdafabe11717670a8523b405e158b995e013568480a6039
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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