第 26074 題：最小頂點覆蓋

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","l0_2","l0_3","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l0_3","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","l0_2","l0_3","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l0_3","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。

• mko-minimum-vertex-cover

逐步推導

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

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

source roles = [[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]]

derived = [[c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]]

source roles = [[c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]], 256]

derived = [[3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 126, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 190, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 252, 253, 254, 255], [[0, 0], [1, 4], [2, 0], [4, 1], [5, 4], [6, 1], [8, 0], [9, 4], [10, 0], [12, 2], [13, 4], [14, 2], [16, 0], [17, 4], [18, 0], [20, 1], [21, 4], [22, 1], [24, 0], [25, 4], [26, 0], [28, 3], [29, 4], [30, 3], [32, 0], [33, 4], [34, 0], [36, 1], [37, 4], [38, 1], [40, 0], [41, 4], [42, 0], [44, 2], [45, 4], [46, 2], [48, 0], [49, 4], [50, 0], [52, 1], [53, 4], [54, 1], [56, 0], [57, 4], [58, 0], [60, 4], [61, 4], [64, 0], [65, 5], [66, 0], [68, 1], [69, 5], [70, 1], [72, 0], [73, 5], [74, 0], [76, 2], [77, 5], [78, 2], [80, 0], [81, 5], [82, 0], [84, 1], [85, 5], [86, 1], [88, 0], [89, 5], [90, 0], [92, 3], [93, 5], [94, 3], [96, 0], [97, 5], [98, 0], [100, 1], [101, 5], [102, 1], [104, 0], [105, 5], [106, 0], [108, 2], [109, 5], [110, 2], [112, 0], [113, 5], [114, 0], [116, 1], [117, 5], [118, 1], [120, 0], [121, 5], [122, 0], [124, 5], [125, 5], [128, 0], [129, 4], [130, 0], [132, 1], [133, 4], [134, 1], [136, 0], [137, 4], [138, 0], [140, 2], [141, 4], [142, 2], [144, 0], [145, 4], [146, 0], [148, 1], [149, 4], [150, 1], [152, 0], [153, 4], [154, 0], [156, 3], [157, 4], [158, 3], [160, 0], [161, 4], [162, 0], [164, 1], [165, 4], [166, 1], [168, 0], [169, 4], [170, 0], [172, 2], [173, 4], [174, 2], [176, 0], [177, 4], [178, 0], [180, 1], [181, 4], [182, 1], [184, 0], [185, 4], [186, 0], [188, 4], [189, 4], [192, 0], [194, 0], [196, 1], [198, 1], [200, 0], [202, 0], [204, 2], [206, 2], [208, 0], [210, 0], [212, 1], [214, 1], [216, 0], [218, 0], [220, 3], [222, 3], [224, 0], [226, 0], [228, 1], [230, 1], [232, 0], [234, 0], [236, 2], [238, 2], [240, 0], [242, 0], [244, 1], [246, 1], [248, 0], [250, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l0_2], [c0, c1, l0_0, l0_2], [c0, c1, l0_1, l0_2], [c0, c1, l0_0, l0_1, l0_2], [c0, c1, l0_3], [c0, c1, l0_0, l0_3], [c0, c1, l0_1, l0_3], [c0, c1, l0_0, l0_1, l0_3], [c0, c1, l0_2, l0_3], [c0, c1, l0_0, l0_2, l0_3], [c0, c1, l0_1, l0_2, l0_3], [c1, l0_0, l0_1, l0_2, l0_3], [c0, c1, l0_0, l0_1, l0_2, l0_3], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l0_2, l1_0], [c0, c1, l0_0, l0_2, l1_0], [c0, c1, l0_1, l0_2, l1_0], [c0, c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l0_3, l1_0], [c0, c1, l0_0, l0_3, l1_0], [c0, c1, l0_1, l0_3, l1_0], [c0, c1, l0_0, l0_1, l0_3, l1_0], [c0, c1, l0_2, l0_3, l1_0], [c0, c1, l0_0, l0_2, l0_3, l1_0], [c0, c1, l0_1, l0_2, l0_3, l1_0], [c1, l0_0, l0_1, l0_2, l0_3, l1_0], [c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l0_2, l1_1], [c0, c1, l0_0, l0_2, l1_1], [c0, c1, l0_1, l0_2, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l0_3, l1_1], [c0, c1, l0_0, l0_3, l1_1], [c0, c1, l0_1, l0_3, l1_1], [c0, c1, l0_0, l0_1, l0_3, l1_1], [c0, c1, l0_2, l0_3, l1_1], [c0, c1, l0_0, l0_2, l0_3, l1_1], [c0, c1, l0_1, l0_2, l0_3, l1_1], [c1, l0_0, l0_1, l0_2, l0_3, l1_1], [c0, c1, l0_0, l0_1, l0_2, l0_3, 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], [c0, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, l0_2, l1_0, l1_1], [c0, c1, l0_2, l1_0, l1_1], [c0, l0_0, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_2, l1_0, l1_1], [c0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_1, l0_2, l1_0, l1_1], [c0, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, l0_3, l1_0, l1_1], [c0, c1, l0_3, l1_0, l1_1], [c0, l0_0, l0_3, l1_0, l1_1], [c0, c1, l0_0, l0_3, l1_0, l1_1], [c0, l0_1, l0_3, l1_0, l1_1], [c0, c1, l0_1, l0_3, l1_0, l1_1], [c0, l0_0, l0_1, l0_3, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_3, l1_0, l1_1], [c0, l0_2, l0_3, l1_0, l1_1], [c0, c1, l0_2, l0_3, l1_0, l1_1], [c0, l0_0, l0_2, l0_3, l1_0, l1_1], [c0, c1, l0_0, l0_2, l0_3, l1_0, l1_1], [c0, l0_1, l0_2, l0_3, l1_0, l1_1], [c0, c1, l0_1, l0_2, l0_3, l1_0, l1_1], [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [c0, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]], [0, 0, 1, 7, 19, 27, 22, 8, 1], 2, [[c0, c1]]]

source roles = [[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]]]

derived = [[[[c0, l0_0], 是], [[c0, l0_1], 是], [[c0, l0_2], 是], [[c0, l0_3], 是], [[c1, l1_0], 是], [[c1, l1_1], 是]], 6, 2, [c0, c1]]

source roles = [[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], 否]]]

derived = [是, 2, [c0, c1], 否]

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, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, 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。