Oriented Edge Sets, Separate Degree Totals and Loop-Free Boundaries
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261108-r07-finite-directed-graph-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
2. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
3. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
4. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
5. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
6. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
7. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
8. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
9. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
10. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
11. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
12. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
13. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
14. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
15. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
16. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
17. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
18. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
19. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
20. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
21. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
22. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
23. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
24. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
25. A Directed Edge Is an Ordered Pair under the Declared Simple Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).
View answer and explanation
(5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)
Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-qualification · candidate
26. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 1, 1], [1, 1, 1, 1, 1], 5, 5, 5, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
27. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 1, 1], [1, 1, 2, 1, 1], 6, 6, 6, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
28. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
29. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 1, 1], [1, 1, 2, 2, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
30. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 1, 1], [1, 1, 1, 1, 2], 6, 6, 6, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
31. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 1, 1], [1, 1, 2, 1, 2], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
32. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 1, 1], [1, 1, 1, 2, 2], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
33. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 1, 1], [1, 1, 2, 2, 2], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
34. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 2, 1], [2, 1, 1, 1, 1], 6, 6, 6, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
35. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 2, 1], [2, 1, 2, 1, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
36. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 2, 1], [2, 1, 1, 2, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
37. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 2, 1], [2, 1, 2, 2, 1], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
38. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 2, 1], [2, 1, 1, 1, 2], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
39. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 2, 1], [2, 1, 2, 1, 2], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
40. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 2, 1], [2, 1, 1, 2, 2], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
41. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1], [2, 1, 2, 2, 2], 9, 9, 9, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
42. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 1, 2], [1, 2, 1, 1, 1], 6, 6, 6, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
43. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 1, 1, 2], [1, 2, 2, 1, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
44. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 1, 1, 2], [1, 2, 1, 2, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
45. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 1, 1, 2], [1, 2, 2, 2, 1], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
46. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], [1, 2, 1, 1, 2], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
47. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2], [1, 2, 2, 1, 2], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
48. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 2, 2, 1, 2], [1, 2, 1, 2, 2], 8, 8, 8, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
49. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([2, 2, 2, 1, 2], [1, 2, 2, 2, 2], 9, 9, 9, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
50. In-Degrees and Out-Degrees Each Sum to the Edge Count
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts).
View answer and explanation
([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, Yes)
Count outgoing and incoming edges separately at each vertex in index order. Both totals equal |E|, not the undirected2|E| degree-sum formula. The exact result, retaining the requested order, is ([1, 1, 1, 2, 2], [2, 2, 1, 1, 1], 7, 7, 7, Yes).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-in-out-degree · candidate
51. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
52. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
53. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
54. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
55. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
56. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
57. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
58. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
59. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
60. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
61. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
62. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
63. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
64. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
65. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
66. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
67. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
68. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
69. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
70. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
71. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
72. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
73. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
74. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
75. Every Successive Step Must Follow Its Own Directed Edge
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention.
View answer and explanation
([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match)
For each consecutive pair report its oriented-edge membership and their conjunction. This is a successive-step audit, not an extra simple-path, distinct-vertex or nonempty-walk certificate. An empty list has no step to refute and makes that conjunction vacuously Yes. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-directed-walk-audit · candidate
76. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 1, 1], [2, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
77. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 1, 1], [2, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
78. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 1, 1], [2, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
79. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 1, 1], [2, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
80. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 1, 1], [2, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
81. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 1, 1], [2, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
82. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 1, 1], [2, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
83. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 1, 1], [2, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
84. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 2, 1], [3, 1, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
85. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
86. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 2, 1], [3, 1, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
87. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 2, 1], [3, 1, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
88. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 2, 1], [3, 1, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
89. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 2, 1], [3, 1, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
90. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
91. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 2, 1], [3, 1, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
92. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 1, 2], [2, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
93. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 1, 1, 2], [2, 2, 2, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
94. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
95. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 1, 1, 2], [2, 2, 2, 2, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
96. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 2, 1, 2], [2, 2, 1, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
97. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 1, 2, 1, 2], [2, 2, 2, 1, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
98. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 1, 2], [2, 2, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
99. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [3, 2, 2, 1, 2], [2, 2, 2, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
100. A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention
Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).
View answer and explanation
([[0, 0]], No, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention)
Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 1, 1, 2, 2], [3, 2, 1, 1, 1], loops_rejected_by_declared_loop_free_simple_digraph_convention).
Reviewed step-by-step derivation
Finite directed graph · qf-mod39-digraph-loop-boundary · candidate
Canonical question JSONL · English JSONL · Validation report