R0049-25430 · q-20261015-advance-20261114-r07-vertex-cover-0055
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"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"superset":["v0","v1","v2","v3","v4","v5","v6"]}。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"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"superset":["v0","v1","v2","v3","v4","v5","v6"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
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。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4, v5]]
操作後
derived = [[v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_complete_simple_undirected_graph_UTF16_then_English_order · 依賴：G001, S001
操作前
source roles = [0, [v0, v1], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S002
操作前
source roles = [1, [v1, v2], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S003
操作前
source roles = [2, [v2, v3], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S004
操作前
source roles = [3, [v3, v4], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S005
操作前
source roles = [4, [v4, v5], [v0, v1, v2, v3, v4, v5]]
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。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S006
操作前
source roles = [[[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是]]]
操作後
derived = [是, null, [v6]]
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。
規則：Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · 依賴：G001, S007
操作前
source roles = [[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v6], [v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5, v6]]
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。
規則：Vcover_empty_graph_conditional_NULL_and_cover_superset_monotonicity · 依賴：G001, S008
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。(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 27002a4a5e9db88b7c3cb7dfa27961a952148949eaa0903cac9628f9fe70b841candidate record SHA-256: 5527994caba3166bf555afae23238c5b1368e4c5793df1c07e7ac44990dde817semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED