第 26092 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-nonminimum-witness；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"]}。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-nonminimum-witness；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"]}。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-nonminimum-witness]

derived = [[candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate]]

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

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

source roles = [[v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 2048]

derived = [[685, 686, 687, 694, 695, 701, 702, 703, 726, 727, 733, 734, 735, 749, 750, 751, 758, 759, 765, 766, 767, 854, 855, 861, 862, 863, 877, 878, 879, 886, 887, 893, 894, 895, 941, 942, 943, 950, 951, 957, 958, 959, 982, 983, 989, 990, 991, 1005, 1006, 1007, 1014, 1015, 1021, 1022, 1023, 1363, 1366, 1367, 1369, 1371, 1373, 1374, 1375, 1385, 1387, 1389, 1390, 1391, 1395, 1398, 1399, 1401, 1403, 1405, 1406, 1407, 1449, 1451, 1453, 1454, 1455, 1459, 1462, 1463, 1465, 1467, 1469, 1470, 1471, 1491, 1494, 1495, 1497, 1499, 1501, 1502, 1503, 1513, 1515, 1517, 1518, 1519, 1523, 1526, 1527, 1529, 1531, 1533, 1534, 1535, 1705, 1707, 1709, 1710, 1711, 1715, 1718, 1719, 1721, 1723, 1725, 1726, 1727, 1747, 1750, 1751, 1753, 1755, 1757, 1758, 1759, 1769, 1771, 1773, 1774, 1775, 1779, 1782, 1783, 1785, 1787, 1789, 1790, 1791, 1875, 1878, 1879, 1881, 1883, 1885, 1886, 1887, 1897, 1899, 1901, 1902, 1903, 1907, 1910, 1911, 1913, 1915, 1917, 1918, 1919, 1961, 1963, 1965, 1966, 1967, 1971, 1974, 1975, 1977, 1979, 1981, 1982, 1983, 2003, 2006, 2007, 2009, 2011, 2013, 2014, 2015, 2025, 2027, 2029, 2030, 2031, 2035, 2038, 2039, 2041, 2043, 2045, 2046, 2047], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [5, 2], [6, 4], [7, 4], [8, 0], [9, 3], [10, 1], [11, 3], [12, 0], [13, 5], [14, 5], [15, 5], [16, 0], [17, 2], [18, 1], [19, 3], [20, 0], [21, 2], [22, 6], [23, 6], [24, 0], [25, 3], [26, 1], [27, 3], [28, 0], [29, 6], [30, 6], [31, 6], [32, 0], [33, 2], [34, 1], [35, 3], [36, 0], [37, 2], [38, 4], [39, 4], [40, 0], [41, 3], [42, 1], [43, 3], [44, 0], [45, 7], [46, 7], [47, 7], [48, 0], [49, 2], [50, 1], [51, 3], [52, 0], [53, 2], [54, 7], [55, 7], [56, 0], [57, 3], [58, 1], [59, 3], [60, 0], [61, 7], [62, 7], [63, 7], [64, 0], [65, 2], [66, 1], [67, 3], [68, 0], [69, 2], [70, 4], [71, 4], [72, 0], [73, 3], [74, 1], [75, 3], [76, 0], [77, 5], [78, 5], [79, 5], [80, 0], [81, 2], [82, 1], [83, 3], [84, 0], [85, 2], [86, 8], [87, 8], [88, 0], [89, 3], [90, 1], [91, 3], [92, 0], [93, 8], 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v3, v4, v5, v6, v7, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v9], [v0, v2, v4, v6, v8, v9], [v0, v1, v2, v4, v6, v8, v9], [v0, v10, v2, v4, v6, v8, v9], [v1, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v1, v3, v4, v6, v8, v9], [v1, v10, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v0, v2, v3, v4, v6, v8, v9], [v0, v1, v2, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v3, v5, v6, v8, v9], [v1, v10, v3, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v0, v2, v3, v5, v6, v8, v9], [v0, v1, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v1, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v2, v4, v5, v6, v8, v9], [v0, v1, v2, v4, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v3, v4, v5, v6, v8, v9], [v1, v10, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v0, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v3, v5, v7, v8, v9], [v1, v10, v3, v5, v7, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v0, v2, v3, v5, v7, v8, v9], [v0, v1, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v1, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v2, v4, v5, v7, v8, v9], [v0, v1, v2, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v3, v4, v5, v7, v8, v9], [v1, v10, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v0, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v2, v4, v6, v7, v8, v9], [v0, v1, v2, v4, v6, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v3, v4, v6, v7, v8, v9], [v1, v10, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v0, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v3, v5, v6, v7, v8, v9], [v1, v10, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v0, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v0, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9]], [0, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1], 6, [[v0, v10, v2, v4, v6, v8], [v1, v10, v2, v4, v6, v8], [v1, v10, v3, v4, v6, v8], [v1, v10, v3, v5, v6, v8], [v1, v10, v3, v5, v7, v8], [v0, v1, v3, v5, v7, v9], [v1, v10, v3, v5, v7, v9], [v0, v2, v3, v5, v7, v9], [v0, v2, v4, v5, v7, v9], [v0, v2, v4, v6, v7, v9], [v0, v2, v4, v6, v8, v9]]]

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

derived = [[[[v0, v1], 是], [[v0, v10], 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]], 11, 6, [v0, v10, v2, v4, v6, v8]]

逐步反向核對

• 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

([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, 11, 6, [v0, v10, v2, v4, v6, v8], 否, explicit_smaller_cover_refutes_minimum_status)

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。