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Question 26008: Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several

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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R0051-26008 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0083

Original question and givens

Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).

  • G001 — Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7] and loop-free duplicate-free undirected edges=[]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=empty_graph. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).

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-empty-multiple-boundary]

After

derived = [[mode, edges, minimum_size, minimum_covers, mode_certificate, boundary]]

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

After

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

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

After

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255], [], [[], [v0], [v1], [v0, v1], [v2], [v0, v2], [v1, v2], [v0, v1, v2], [v3], [v0, v3], [v1, v3], [v0, v1, v3], [v2, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v4], [v0, v4], [v1, v4], [v0, v1, v4], [v2, v4], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v3, v4], [v0, v3, v4], [v1, v3, v4], [v0, v1, v3, v4], [v2, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v5], [v0, v5], [v1, v5], [v0, v1, v5], [v2, v5], [v0, v2, v5], [v1, v2, v5], [v0, v1, v2, v5], [v3, v5], [v0, v3, v5], [v1, v3, v5], [v0, v1, v3, v5], [v2, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v4, v5], [v0, v4, v5], [v1, v4, v5], [v0, v1, v4, v5], [v2, v4, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v3, v4, v5], [v0, v3, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v2, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v6], [v0, v6], [v1, v6], [v0, v1, v6], [v2, v6], [v0, v2, v6], [v1, v2, v6], [v0, v1, v2, v6], [v3, v6], [v0, v3, v6], [v1, v3, v6], [v0, v1, v3, v6], [v2, v3, v6], [v0, v2, v3, v6], [v1, v2, v3, v6], [v0, v1, v2, v3, v6], [v4, v6], [v0, v4, v6], [v1, v4, v6], [v0, v1, v4, v6], [v2, v4, v6], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v3, v4, v6], [v0, v3, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v2, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v5, v6], [v0, v5, v6], [v1, v5, v6], [v0, v1, v5, v6], [v2, v5, v6], [v0, v2, v5, v6], [v1, v2, v5, v6], [v0, v1, v2, v5, v6], [v3, v5, v6], [v0, v3, v5, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v2, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v4, v5, v6], [v0, v4, v5, v6], [v1, v4, v5, v6], [v0, v1, v4, v5, v6], [v2, v4, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v3, v4, v5, v6], [v0, v3, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v2, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v7], [v0, v7], [v1, v7], [v0, v1, v7], [v2, v7], [v0, v2, v7], [v1, v2, v7], [v0, v1, v2, v7], [v3, v7], [v0, v3, v7], [v1, v3, v7], [v0, v1, v3, v7], [v2, v3, v7], [v0, v2, v3, v7], [v1, v2, v3, v7], [v0, v1, v2, v3, v7], [v4, v7], [v0, v4, v7], [v1, v4, v7], [v0, v1, v4, v7], [v2, v4, v7], [v0, v2, v4, v7], [v1, v2, v4, v7], [v0, v1, v2, v4, v7], [v3, v4, v7], [v0, v3, v4, v7], [v1, v3, v4, v7], [v0, v1, v3, v4, v7], [v2, v3, v4, v7], [v0, v2, v3, v4, v7], [v1, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v5, v7], [v0, v5, v7], [v1, v5, v7], [v0, v1, v5, v7], [v2, v5, v7], [v0, v2, v5, v7], [v1, v2, v5, v7], [v0, v1, v2, v5, v7], [v3, v5, v7], [v0, v3, v5, v7], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v2, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v4, v5, v7], [v0, v4, v5, v7], [v1, v4, v5, v7], [v0, v1, v4, v5, v7], [v2, v4, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v3, v4, v5, v7], [v0, v3, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v2, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v6, v7], [v0, v6, v7], [v1, v6, v7], [v0, v1, v6, v7], [v2, v6, v7], [v0, v2, v6, v7], [v1, v2, v6, v7], [v0, v1, v2, v6, v7], [v3, v6, v7], [v0, v3, v6, v7], [v1, v3, v6, v7], [v0, v1, v3, v6, v7], [v2, v3, v6, v7], [v0, v2, v3, v6, v7], [v1, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v4, v6, v7], [v0, v4, v6, v7], [v1, v4, v6, v7], [v0, v1, v4, v6, v7], [v2, v4, v6, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v3, v4, v6, v7], [v0, v3, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v2, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v5, v6, v7], [v0, v5, v6, v7], [v1, v5, v6, v7], [v0, v1, v5, v6, v7], [v2, v5, v6, v7], [v0, v2, v5, v6, v7], [v1, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v3, v5, v6, v7], [v0, v3, v5, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v2, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v4, v5, v6, v7], [v0, v4, v5, v6, v7], [v1, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v2, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v3, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v2, 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]], [1, 8, 28, 56, 70, 56, 28, 8, 1], 0, [[]]]

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

After

derived = [0, [[]], Yes]

Conditions: none

empty_graph means edgeless,not an empty vertex carrier;the unique minimum is empty.multiple_minima retains all of at least two solutions,not one witness as uniqueness.No Konig,weighted or directed extension.

Step identifiers and dependencies

Rule: Mincover_original_empty_OR_multiple_boundary_not_uniqueness_guess · Uses: G001, S003

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

(empty_graph, [], 0, [[]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)

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: 0d35260f2dec8c4de03912bd61478bd760ea447bbdbb7a71b33235f78990a1f7
  • candidate record SHA-256: fc4cd44331dcb64d7b1b1b062b3b9489e183aaa08da94323bb5c51139028adb1
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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