第 20830 題：有限有向圖

原題與已知條件

完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-in-out-degree；資料={"n":5,"edges":[[0,1],[1,2],[2,3],[2,4],[3,4],[4,0]]}。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-finite-directed-graph

逐步推導

source operation=digraph-in-out-degree;original snapshot and mathematics retained

explicit ordered source output fields=[out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts]

ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]

all endpoints in range and same-direction arcs unique;actual loops=[];main loop-free directed-set convention qualified

digraph vertex0 exact outgoing count := sum(1)

digraph vertex0 exact outgoing count = 1

digraph vertex0 exact incoming count := sum(1)

digraph vertex0 exact incoming count = 1

ORIGINAL vertex=0;ALL outgoing original-arc indices=[0];ALL incoming indices=[5]

outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex1 exact outgoing count := sum(1)

digraph vertex1 exact outgoing count = 1

digraph vertex1 exact incoming count := sum(1)

digraph vertex1 exact incoming count = 1

ORIGINAL vertex=1;ALL outgoing original-arc indices=[1];ALL incoming indices=[0]

outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex2 exact outgoing count := sum(1, 1)

digraph vertex2 exact outgoing count = 2

digraph vertex2 exact incoming count := sum(1)

digraph vertex2 exact incoming count = 1

ORIGINAL vertex=2;ALL outgoing original-arc indices=[2, 3];ALL incoming indices=[1]

outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex3 exact outgoing count := sum(1)

digraph vertex3 exact outgoing count = 1

digraph vertex3 exact incoming count := sum(1)

digraph vertex3 exact incoming count = 1

ORIGINAL vertex=3;ALL outgoing original-arc indices=[4];ALL incoming indices=[2]

outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count

digraph vertex4 exact outgoing count := sum(1)

digraph vertex4 exact outgoing count = 1

digraph vertex4 exact incoming count := sum(1, 1)

digraph vertex4 exact incoming count = 2

ORIGINAL vertex=4;ALL outgoing original-arc indices=[5];ALL incoming indices=[3, 4]

outdegree=1;indegree=2;a loop at this vertex contributes ONCE to EACH separate count

digraph complete outgoing-degree sum := sum(1, 1, 2, 1, 1)

digraph complete outgoing-degree sum = 6

digraph complete incoming-degree sum := sum(1, 1, 1, 1, 2)

digraph complete incoming-degree sum = 6

ALL vertex-order outdegrees=[1, 1, 2, 1, 1];indegrees=[1, 1, 1, 1, 2];complete directed arc count=6

each arc counted once at its source and once at its target;SUM(out)=6=|E| and SUM(in)=6=|E|;the SEPARATE counting rule is |E|,not an imported undirected2|E| rule (numbers may coincide when E=0)

逐步反向核對

• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 digraph complete incoming-degree sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 digraph complete outgoing-degree sum，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 digraph vertex4 exact incoming count，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 digraph vertex4 exact outgoing count，不以小數近似替代。
• 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 — 以精確有理數重算 digraph vertex3 exact incoming count，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 digraph vertex3 exact outgoing count，不以小數近似替代。
• 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 — 以精確有理數重算 digraph vertex2 exact incoming count，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 digraph vertex2 exact outgoing count，不以小數近似替代。
• 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 — 以精確有理數重算 digraph vertex1 exact incoming count，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 digraph vertex1 exact outgoing count，不以小數近似替代。
• 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 — 以精確有理數重算 digraph vertex0 exact incoming count，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 digraph vertex0 exact outgoing count，不以小數近似替代。
• 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

([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, 是)

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。