# Question 26009: Edgeless Graphs Have the Unique Empty Minimum Cover; Other Graphs May Have Several [Return to catalogue page 131](https://ocme.evemisslab.com/en/questions/page/131/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations/#q-20261015-advance-20261115-r06-minimum-vertex-cover-0084) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0051-26009 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0084 ## Original question and givens Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8] 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, v8] 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. - [mko-minimum-vertex-cover](https://ocme.evemisslab.com/en/mko/mko-minimum-vertex-cover/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [], null] After derived = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [], 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 dependenciesRule: Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · Uses: G001, S001 S003 · impliesBefore source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [], 512] 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, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 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[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, 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v6, v8], [v0, v2, v3, v4, v6, v8], [v1, v2, v3, v4, v6, v8], [v0, v1, v2, v3, v4, v6, v8], [v5, v6, v8], [v0, v5, v6, v8], [v1, v5, v6, v8], [v0, v1, v5, v6, v8], [v2, v5, v6, v8], [v0, v2, v5, v6, v8], [v1, v2, v5, v6, v8], [v0, v1, v2, v5, v6, v8], [v3, v5, v6, v8], [v0, v3, v5, v6, v8], [v1, v3, v5, v6, v8], [v0, v1, v3, v5, v6, v8], [v2, v3, v5, v6, v8], [v0, v2, v3, v5, v6, v8], [v1, v2, v3, v5, v6, v8], [v0, v1, v2, v3, v5, v6, v8], [v4, v5, v6, v8], [v0, v4, v5, v6, v8], [v1, v4, v5, v6, v8], [v0, v1, v4, v5, v6, v8], [v2, v4, v5, v6, v8], [v0, v2, v4, v5, v6, v8], [v1, v2, v4, v5, v6, v8], [v0, v1, v2, v4, v5, v6, v8], [v3, v4, v5, v6, v8], [v0, v3, v4, v5, v6, v8], [v1, v3, v4, v5, v6, v8], [v0, v1, v3, v4, v5, v6, v8], [v2, v3, v4, v5, v6, v8], [v0, v2, v3, v4, v5, v6, v8], [v1, v2, v3, v4, v5, v6, v8], [v0, v1, v2, v3, v4, v5, v6, v8], [v7, v8], [v0, v7, v8], [v1, v7, v8], [v0, v1, v7, v8], [v2, v7, v8], [v0, v2, v7, v8], [v1, v2, v7, v8], [v0, v1, v2, v7, v8], [v3, v7, v8], [v0, v3, v7, v8], [v1, v3, v7, v8], [v0, v1, v3, v7, v8], [v2, v3, v7, v8], [v0, v2, v3, v7, v8], [v1, v2, v3, v7, v8], [v0, v1, v2, v3, v7, v8], [v4, v7, v8], [v0, v4, v7, v8], [v1, v4, v7, v8], [v0, v1, v4, v7, v8], [v2, v4, v7, v8], [v0, v2, v4, v7, v8], [v1, v2, v4, v7, v8], [v0, v1, v2, v4, v7, v8], [v3, v4, v7, v8], [v0, v3, v4, v7, v8], [v1, v3, v4, v7, v8], [v0, v1, v3, v4, v7, v8], [v2, v3, v4, v7, v8], [v0, v2, v3, v4, v7, v8], [v1, v2, v3, v4, v7, v8], [v0, v1, v2, v3, v4, v7, v8], [v5, v7, v8], [v0, v5, v7, v8], [v1, v5, v7, v8], [v0, v1, v5, v7, v8], [v2, v5, v7, v8], [v0, v2, v5, v7, v8], [v1, v2, v5, v7, v8], [v0, v1, v2, v5, v7, v8], [v3, v5, v7, v8], [v0, v3, v5, v7, v8], [v1, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v8], [v2, v3, v5, v7, v8], [v0, v2, v3, v5, v7, v8], [v1, v2, v3, v5, v7, v8], [v0, v1, v2, v3, v5, v7, v8], [v4, v5, v7, v8], [v0, v4, v5, v7, v8], [v1, v4, v5, v7, v8], [v0, v1, v4, v5, v7, v8], [v2, v4, v5, v7, v8], [v0, v2, v4, v5, v7, v8], [v1, v2, v4, v5, v7, v8], [v0, v1, v2, v4, v5, v7, v8], [v3, v4, v5, v7, v8], [v0, v3, v4, v5, v7, v8], [v1, v3, v4, v5, v7, v8], [v0, v1, v3, v4, v5, v7, v8], [v2, v3, v4, v5, v7, v8], [v0, v2, v3, v4, v5, v7, v8], [v1, v2, v3, v4, v5, v7, v8], [v0, v1, v2, v3, v4, v5, v7, v8], [v6, v7, v8], [v0, v6, v7, v8], [v1, v6, v7, v8], [v0, v1, v6, v7, v8], [v2, v6, v7, v8], [v0, v2, v6, v7, v8], [v1, v2, v6, v7, v8], [v0, v1, v2, v6, v7, v8], [v3, v6, v7, v8], [v0, v3, v6, v7, v8], [v1, v3, v6, v7, v8], [v0, v1, v3, v6, v7, v8], [v2, v3, v6, v7, v8], [v0, v2, v3, v6, v7, v8], [v1, v2, v3, v6, v7, v8], [v0, v1, v2, v3, v6, v7, v8], [v4, v6, v7, v8], [v0, v4, v6, v7, v8], [v1, v4, v6, v7, v8], [v0, v1, v4, v6, v7, v8], [v2, v4, v6, v7, v8], [v0, v2, v4, v6, v7, v8], [v1, v2, v4, v6, v7, v8], [v0, v1, v2, v4, v6, v7, v8], [v3, v4, v6, v7, v8], [v0, v3, v4, v6, v7, v8], [v1, v3, v4, v6, v7, v8], [v0, v1, v3, v4, v6, v7, v8], [v2, v3, v4, v6, v7, v8], [v0, v2, v3, v4, v6, v7, v8], [v1, v2, v3, v4, v6, v7, v8], [v0, v1, v2, v3, v4, v6, v7, v8], [v5, v6, v7, v8], [v0, v5, v6, v7, v8], [v1, v5, v6, v7, v8], [v0, v1, v5, v6, v7, v8], [v2, v5, v6, v7, v8], [v0, v2, v5, v6, v7, v8], [v1, v2, v5, v6, v7, v8], [v0, v1, v2, v5, v6, v7, v8], [v3, v5, v6, v7, v8], [v0, v3, v5, v6, v7, v8], [v1, v3, v5, v6, v7, v8], [v0, v1, v3, v5, v6, v7, v8], [v2, v3, v5, v6, v7, v8], [v0, v2, v3, v5, v6, v7, v8], [v1, v2, v3, v5, v6, v7, v8], [v0, v1, v2, v3, v5, v6, v7, v8], [v4, v5, v6, v7, v8], [v0, v4, v5, v6, v7, v8], [v1, v4, v5, v6, v7, v8], [v0, v1, v4, v5, v6, v7, v8], [v2, v4, v5, v6, v7, v8], [v0, v2, v4, v5, v6, v7, v8], [v1, v2, v4, v5, v6, v7, v8], [v0, v1, v2, v4, v5, v6, v7, v8], [v3, v4, v5, v6, v7, v8], [v0, v3, v4, v5, v6, v7, v8], [v1, v3, v4, v5, v6, v7, v8], [v0, v1, v3, v4, v5, v6, v7, v8], [v2, v3, v4, v5, v6, v7, v8], [v0, v2, v3, v4, v5, v6, v7, v8], [v1, v2, v3, v4, v5, v6, v7, v8], [v0, v1, v2, v3, v4, v5, v6, v7, v8]], [1, 9, 36, 84, 126, 126, 84, 36, 9, 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 dependenciesRule: Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · Uses: G001, S002 S004 · impliesBefore source roles = [empty_graph, [v0, v1, v2, v3, v4, v5, v6, v7, v8], []] 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 dependenciesRule: 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: ccab99d121cea27c03fd36feb962661744c320a890d6f6d1a857d9317077ebb2 - candidate record SHA-256: bcbc1c3bcdaad3bcaa01c92b8b0e32ac2166138d129c4e21739b43ea087790ea - semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3 - question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r06-minimum-vertex-cover-0084.json)