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Question 26069: 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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R0051-26069 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0069

Original question and givens

Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]. 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, c1, l1_0, l1_1, l1_2] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]. 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, c1, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]]

After

derived = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]]

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, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]], 256]

After

derived = [[3, 7, 11, 15, 19, 23, 27, 30, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 94, 95, 99, 103, 107, 111, 115, 119, 123, 126, 127, 131, 135, 139, 143, 147, 151, 155, 158, 159, 163, 167, 171, 175, 179, 183, 187, 190, 191, 195, 199, 203, 207, 211, 215, 219, 222, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 252, 253, 254, 255], [[0, 0], [1, 3], [2, 0], [4, 1], [5, 3], [6, 1], [8, 0], [9, 3], [10, 0], [12, 2], [13, 3], [14, 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, 4], [34, 0], [36, 1], [37, 4], [38, 1], [40, 0], [41, 4], [42, 0], [44, 2], [45, 4], [46, 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, 3], [66, 0], [68, 1], [69, 3], [70, 1], [72, 0], [73, 3], [74, 0], [76, 2], [77, 3], [78, 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, 5], [98, 0], [100, 1], [101, 5], [102, 1], [104, 0], [105, 5], [106, 0], [108, 2], [109, 5], [110, 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, 3], [130, 0], [132, 1], [133, 3], [134, 1], [136, 0], [137, 3], [138, 0], [140, 2], [141, 3], [142, 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, 4], [162, 0], [164, 1], [165, 4], [166, 1], [168, 0], [169, 4], [170, 0], [172, 2], [173, 4], [174, 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, 3], [194, 0], [196, 1], [197, 3], [198, 1], [200, 0], [201, 3], [202, 0], [204, 2], [205, 3], [206, 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], [226, 0], [228, 1], [230, 1], [232, 0], [234, 0], [236, 2], [238, 2], [240, 0], [242, 0], [244, 1], [246, 1], [248, 0], [250, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l0_2], [c0, c1, l0_0, l0_2], [c0, c1, l0_1, l0_2], [c1, l0_0, l0_1, l0_2], [c0, c1, l0_0, l0_1, l0_2], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l0_2, l1_0], [c0, c1, l0_0, l0_2, l1_0], [c0, c1, l0_1, l0_2, l1_0], [c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l0_2, l1_1], [c0, c1, l0_0, l0_2, l1_1], [c0, c1, l0_1, l0_2, l1_1], [c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_2, l1_0, l1_1], [c0, c1, l0_1, l0_2, l1_0, l1_1], [c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l1_2], [c0, c1, l0_0, l1_2], [c0, c1, l0_1, l1_2], [c0, c1, l0_0, l0_1, l1_2], [c0, c1, l0_2, l1_2], [c0, c1, l0_0, l0_2, l1_2], [c0, c1, l0_1, l0_2, l1_2], [c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l1_0, l1_2], [c0, c1, l0_0, l1_0, l1_2], [c0, c1, l0_1, l1_0, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_2], [c0, c1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_1, l0_2, 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], [c0, c1, l0_0, l0_1, l1_1, l1_2], [c0, c1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, l1_0, l1_1, l1_2], [c0, c1, l1_0, l1_1, l1_2], [c0, l0_0, l1_0, l1_1, l1_2], [c0, c1, l0_0, l1_0, l1_1, l1_2], [c0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_1, l1_0, l1_1, l1_2], [c0, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_2, l1_0, l1_1, l1_2], [c0, l0_0, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2], [c0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]], [0, 0, 1, 6, 17, 26, 22, 8, 1], 2, [[c0, c1]]]

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, l1_0, l1_1, l1_2], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]]]

After

derived = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes]], 6, 2, [c0, c1]]

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, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]]]

After

derived = [Yes, 2, [c0, c1], 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, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 3feadc1f3a8de7f74dc5405bd47f756b46ac29e699f053f727eedde6b5a66b8b
  • candidate record SHA-256: ce43b6a4e55a01377cdcdc77401fd18632adb7c83feb72cb2f593b011fcc7fc9
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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