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第 26061 題：最小頂點覆蓋

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  1. 原題與已知條件
  2. 定義與適用範圍
  3. 逐步推導
  4. 逐步反向核對
  5. 結論
  6. 本題覆核與來源綁定

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← 逐題詳解目錄 · 原題批次

R0051-26061 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0061

原題與已知條件

bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l1_0","l1_1"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

  • G001 — bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l1_0","l1_1"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

逐步推導

S001 · implies

操作前

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

操作後

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

條件：無

本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。

步驟識別與依賴

規則：source_ordered_fields_without_JS · 依賴：G001

S002 · implies

操作前

source roles = [[c0, l0_0, l0_1, c1, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]], [l0_0, l0_1, l1_0, l1_1]]

操作後

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

條件：無

非空carrier至多12；端點UTF16排序後edge族English排序，loop/反向duplicate拒絕。未提供candidate時保持null，不擅補空candidate。

步驟識別與依賴

規則：Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · 依賴：G001, S001

S003 · implies

操作前

source roles = [[c0, c1, l0_0, l0_1, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]], 64]

操作後

derived = [[3, 7, 11, 14, 15, 19, 23, 27, 30, 31, 35, 39, 43, 46, 47, 49, 51, 53, 55, 57, 59, 60, 61, 62, 63], [[0, 0], [1, 2], [2, 0], [4, 1], [5, 2], [6, 1], [8, 0], [9, 2], [10, 0], [12, 2], [13, 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, 2], [34, 0], [36, 1], [37, 2], [38, 1], [40, 0], [41, 2], [42, 0], [44, 2], [45, 2], [48, 0], [50, 0], [52, 1], [54, 1], [56, 0], [58, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c1, l0_0, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c1, l0_0, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c1, l0_0, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, l1_0, l1_1], [c0, c1, l1_0, l1_1], [c0, l0_0, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, l0_1, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [l0_0, l0_1, l1_0, l1_1], [c0, l0_0, l0_1, l1_0, l1_1], [c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1]], [0, 0, 1, 6, 11, 6, 1], 2, [[c0, c1]]]

條件：無

逐全部mask核所有edges；每個失敗mask保首uncovered edge index反例，通過者全列source順序。較小size計數0給下界，實際最小解給上界；全部ties依numeric mask保留，不改English/大小序。

步驟識別與依賴

規則：Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · 依賴：G001, S002

S004 · implies

操作前

source roles = [[l0_0, l0_1, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]]]

操作後

derived = [[[[c0, l0_0], 是], [[c0, l0_1], 是], [[c1, l1_0], 是], [[c1, l1_1], 是]], 4, 2, [c0, c1]]

條件：無

先核candidate真cover；較小解取完整source清單的首minimum，不用任意小集合代替。

步驟識別與依賴

規則：Mincover_actual_candidate_complete_cover_and_smaller_witness · 依賴：G001, S003

S005 · implies

操作前

source roles = [[l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]]]

操作後

derived = [是, 2, [c0, c1], 否]

條件：無

逐刪都失cover，所以inclusion-minimal；但完整枚舉已有嚴格更小cover，所以不是minimum。cover對superset單調使逐刪可驗全部proper subset責任，不等同最大／最優大小。

步驟識別與依賴

規則：Mincover_ALL_single_deletions_minimal_BUT_smaller_cover_exists · 依賴：G001, S004

逐步反向核對

  • S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。

來源雜湊與覆核憑證
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d47b71f0ec5ba54b4ad80c7ea76f705ed26a704cf99b849bebb3998f1acb9164
  • candidate record SHA-256: f2ccfe26f5669233dfed7063afe08e07d7a8606fde8a5ed0e0317ec9b2d13980
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON