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第 25435 題：頂點覆蓋

本頁導覽
  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

R0049-25435 · q-20261015-advance-20261114-r07-vertex-cover-0060

原題與已知條件

有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

  • G001 — 有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"]}。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 = [vcover-empty-isolated-superset]

操作後

derived = [[edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate]]

條件：無

本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，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 = [[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]]

操作後

derived = [[v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

條件：無

carrier非空且至多12，edge端點依JS UTF16<排，再按來源English canonical排序；loop或反向重複edge拒絕，不把圖改成有向或edge cover。

步驟識別與依賴

規則：Vcover_complete_simple_undirected_graph_UTF16_then_English_order · 依賴：G001, S001

S003 · implies

操作前

source roles = [0, [v0, v1], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S002

S004 · implies

操作前

source roles = [1, [v1, v2], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S003

S005 · implies

操作前

source roles = [2, [v10, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S004

S006 · implies

操作前

source roles = [3, [v2, v3], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S005

S007 · implies

操作前

source roles = [4, [v3, v4], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S006

S008 · implies

操作前

source roles = [5, [v4, v5], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S007

S009 · implies

操作前

source roles = [6, [v5, v6], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S008

S010 · implies

操作前

source roles = [7, [v6, v7], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S009

S011 · implies

操作前

source roles = [8, [v7, v8], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S010

S012 · implies

操作前

source roles = [9, [v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [[是, 是], 是]

條件：無

這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。

步驟識別與依賴

規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S011

S013 · implies

操作前

source roles = [[[[v0, v1], 是], [[v1, v2], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]]]

操作後

derived = [是, null, [v11]]

條件：無

全邊合取，首uncovered edge保來源canonical次序；沒有則null，不用False代null。isolated點沒有責任edge，故可選可不選。

步驟識別與依賴

規則：Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · 依賴：G001, S012

S014 · implies

操作前

source roles = [[[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]]

操作後

derived = [否, null, 是]

條件：無

空edge族時空cover合法；非空edge時此conditional欄是null。若C已cover且C⊆S，每條edge原選中endpoint仍在S，故S也cover；superset可等於C。

步驟識別與依賴

規則：Vcover_empty_graph_conditional_NULL_and_cover_superset_monotonicity · 依賴：G001, S013

逐步反向核對

  • S014 · 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。
  • S013 · 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。
  • S012 · 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。
  • S011 · 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。
  • S010 · 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。
  • S009 · 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。
  • S008 · 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。
  • S007 · 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。
  • S006 · 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。
  • 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

(否, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

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: bb8651d855088319fe45f5d6c104446086b16b2a034cc3e1ac8f66e37ed41fa4
  • candidate record SHA-256: 877f49ceaa99f9b02d62a1d3f71ba3b2058da09d5548dcd9bec48fa3f86878f6
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON