Simple Undirected Graph Inventories and Induced Subsets
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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-20261012-advance-20261017-r05-graph-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(0, [], [], [], 0, 0)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], [], 0, 0).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
2. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(1, [], [[No]], [0], 0, 0)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[No]], [0], 0, 0).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
3. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], [[No, No, No, No], [No, No, Yes, Yes], [No, Yes, No, Yes], [No, Yes, Yes, No]], [0, 2, 2, 2], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[No, No, No, No], [No, No, Yes, Yes], [No, Yes, No, Yes], [No, Yes, Yes, No]], [0, 2, 2, 2], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
4. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(4, [[0, 1], [2, 3]], [[No, Yes, No, No], [Yes, No, No, No], [No, No, No, Yes], [No, No, Yes, No]], [1, 1, 1, 1], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[No, Yes, No, No], [Yes, No, No, No], [No, No, No, Yes], [No, No, Yes, No]], [1, 1, 1, 1], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
5. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(4, [[0, 1], [1, 2], [2, 3]], [[No, Yes, No, No], [Yes, No, Yes, No], [No, Yes, No, Yes], [No, No, Yes, No]], [1, 2, 2, 1], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[No, Yes, No, No], [Yes, No, Yes, No], [No, Yes, No, Yes], [No, No, Yes, No]], [1, 2, 2, 1], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
6. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(4, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes], [Yes, No, No, No], [Yes, No, No, No], [Yes, No, No, No]], [3, 1, 1, 1], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes], [Yes, No, No, No], [Yes, No, No, No], [Yes, No, No, No]], [3, 1, 1, 1], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
7. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1]], [[No, Yes, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 0, 0, 0], 2, 2)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[No, Yes, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 0, 0, 0], 2, 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
8. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 2]], [[No, No, Yes, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 0, 1, 0, 0], 2, 2)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[No, No, Yes, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 0, 1, 0, 0], 2, 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
9. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 2]], [[No, Yes, Yes, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 1, 1, 0, 0], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[No, Yes, Yes, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 1, 1, 0, 0], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
10. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 3]], [[No, No, No, Yes, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [1, 0, 0, 1, 0], 2, 2)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[No, No, No, Yes, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [1, 0, 0, 1, 0], 2, 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
11. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 3]], [[No, Yes, No, Yes, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 1, 0, 1, 0], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[No, Yes, No, Yes, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 1, 0, 1, 0], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
12. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 2], [0, 3]], [[No, No, Yes, Yes, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 0, 1, 1, 0], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[No, No, Yes, Yes, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [2, 0, 1, 1, 0], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
13. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [3, 1, 1, 1, 0], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[No, Yes, Yes, Yes, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No]], [3, 1, 1, 1, 0], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
14. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 4]], [[No, No, No, No, Yes], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [1, 0, 0, 0, 1], 2, 2)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[No, No, No, No, Yes], [No, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [1, 0, 0, 0, 1], 2, 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
15. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 4]], [[No, Yes, No, No, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 1, 0, 0, 1], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[No, Yes, No, No, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 1, 0, 0, 1], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
16. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 2], [0, 4]], [[No, No, Yes, No, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 1, 0, 1], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[No, No, Yes, No, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 1, 0, 1], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
17. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 4]], [[No, Yes, Yes, No, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 1, 0, 1], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[No, Yes, Yes, No, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 1, 0, 1], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
18. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 3], [0, 4]], [[No, No, No, Yes, Yes], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 0, 1, 1], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[No, No, No, Yes, Yes], [No, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [2, 0, 0, 1, 1], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
19. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 3], [0, 4]], [[No, Yes, No, Yes, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 0, 1, 1], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[No, Yes, No, Yes, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 0, 1, 1], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
20. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 2], [0, 3], [0, 4]], [[No, No, Yes, Yes, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 0, 1, 1, 1], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[No, No, Yes, Yes, Yes], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 0, 1, 1, 1], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
21. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[No, Yes, Yes, Yes, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [4, 1, 1, 1, 1], 8, 8)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[No, Yes, Yes, Yes, Yes], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [4, 1, 1, 1, 1], 8, 8).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
22. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[1, 2]], [[No, No, No, No, No], [No, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [0, 1, 1, 0, 0], 2, 2)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[No, No, No, No, No], [No, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [0, 1, 1, 0, 0], 2, 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
23. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [1, 2]], [[No, Yes, No, No, No], [Yes, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 2, 1, 0, 0], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[No, Yes, No, No, No], [Yes, No, Yes, No, No], [No, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 2, 1, 0, 0], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
24. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 2], [1, 2]], [[No, No, Yes, No, No], [No, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 2, 0, 0], 4, 4)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[No, No, Yes, No, No], [No, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [1, 1, 2, 0, 0], 4, 4).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
25. Complete Adjacency Matrix, Degrees, and Handshake Counts
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2]], [[No, Yes, Yes, No, No], [Yes, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 2, 2, 0, 0], 6, 6)
Read the complete symmetric adjacency relation and count each distinct undirected edge once; this is not a connectivity or tree test. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[No, Yes, Yes, No, No], [Yes, No, Yes, No, No], [Yes, Yes, No, No, No], [No, No, No, No, No], [No, No, No, No, No]], [2, 2, 2, 0, 0], 6, 6).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-adjacency-degrees · candidate
26. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(0, [], [], [], [])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], [], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
27. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(1, [], [[]], [[]], [0])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[]], [[]], [0]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
28. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[], [2, 3], [1, 3], [1, 2]], [[1, 2, 3], [0], [0], [0]], [0]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
29. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], [])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[1], [0], [3], [2]], [[2, 3], [2, 3], [0, 1], [0, 1]], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
30. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], [])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[1], [0, 2], [1, 3], [2]], [[2, 3], [3], [0], [0, 1]], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
31. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], [])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0]], [[], [2, 3], [1, 3], [1, 2]], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
32. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[1], [0], [], [], []], [[2, 3, 4], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [2, 3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
33. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[2], [], [0], [], []], [[1, 3, 4], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [1, 3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
34. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[1, 2], [0], [0], [], []], [[3, 4], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
35. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[3], [], [], [0], []], [[1, 2, 4], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 2, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
36. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[1, 3], [0], [], [0], []], [[2, 4], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [2, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
37. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[2, 3], [], [0], [0], []], [[1, 4], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [1, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
38. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[1, 2, 3], [0], [0], [0], []], [[4], [2, 3, 4], [1, 3, 4], [1, 2, 4], [0, 1, 2, 3]], [4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
39. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[4], [], [], [], [0]], [[1, 2, 3], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 2, 3]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
40. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[1, 4], [0], [], [], [0]], [[2, 3], [2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [2, 3]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
41. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[2, 4], [], [0], [], [0]], [[1, 3], [0, 2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [1, 3]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
42. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[1, 2, 4], [0], [0], [], [0]], [[3], [2, 3, 4], [1, 3, 4], [0, 1, 2, 4], [1, 2, 3]], [3]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
43. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[3, 4], [], [], [0], [0]], [[1, 2], [0, 2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [1, 2]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
44. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[1, 3, 4], [0], [], [0], [0]], [[2], [2, 3, 4], [0, 1, 3, 4], [1, 2, 4], [1, 2, 3]], [2]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
45. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[2, 3, 4], [], [0], [0], [0]], [[1], [0, 2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [1]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
46. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], [])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[1, 2, 3, 4], [0], [0], [0], [0]], [[], [2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
47. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[], [2], [1], [], []], [[1, 2, 3, 4], [0, 3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [0, 3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
48. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[1], [0, 2], [1], [], []], [[2, 3, 4], [3, 4], [0, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
49. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
50. Every Vertex Neighbor Partition and Isolated Vertices
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])
Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[1, 2], [0, 2], [0, 1], [], []], [[3, 4], [3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-complete-neighbor-partitions · candidate
51. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[1, 0]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1]], [0, 1], removed, [], [1, 1], [0, 0], [-1, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 1], removed, [], [1, 1], [0, 0], [-1, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
52. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 1]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([], [0, 1], added, [[0, 1]], [0, 0], [1, 1], [1, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 1], added, [[0, 1]], [0, 0], [1, 1], [1, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
53. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[2, 0]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [1, 2]], [0, 2], added, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2], added, [[0, 1], [0, 2], [1, 2]], [1, 2, 1], [2, 2, 2], [1, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
54. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 1]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2]], [0, 1], removed, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 1], removed, [[0, 2]], [2, 1, 1], [1, 0, 1], [-1, -1, 0], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
55. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1]], [0, 4], added, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 4], added, [[0, 1], [0, 4]], [1, 1, 0, 0, 0], [2, 1, 0, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
56. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 2]], [0, 4], added, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2]], [0, 4], added, [[0, 2], [0, 4]], [1, 0, 1, 0, 0], [2, 0, 1, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
57. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4]], [2, 1, 1, 0, 0], [3, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
58. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 3]], [0, 4], added, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3]], [0, 4], added, [[0, 3], [0, 4]], [1, 0, 0, 1, 0], [2, 0, 0, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
59. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 3]], [0, 4], added, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3]], [0, 4], added, [[0, 1], [0, 3], [0, 4]], [2, 1, 0, 1, 0], [3, 1, 0, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
60. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 2], [0, 3]], [0, 4], added, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3]], [0, 4], added, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
61. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2], [0, 3]], [0, 4], added, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3]], [0, 4], added, [[0, 1], [0, 2], [0, 3], [0, 4]], [3, 1, 1, 1, 0], [4, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
62. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 4]], [0, 4], removed, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 4]], [0, 4], removed, [], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
63. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 4]], [0, 4], removed, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 4]], [0, 4], removed, [[0, 1]], [2, 1, 0, 0, 1], [1, 1, 0, 0, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
64. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 2], [0, 4]], [0, 4], removed, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 4]], [0, 4], removed, [[0, 2]], [2, 0, 1, 0, 1], [1, 0, 1, 0, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
65. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2], [0, 4]], [0, 4], removed, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 4]], [0, 4], removed, [[0, 1], [0, 2]], [3, 1, 1, 0, 1], [2, 1, 1, 0, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
66. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 3], [0, 4]], [0, 4], removed, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4]], [0, 4], removed, [[0, 3]], [2, 0, 0, 1, 1], [1, 0, 0, 1, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
67. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 3]], [3, 1, 0, 1, 1], [2, 1, 0, 1, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
68. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
69. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 1], [0, 2], [0, 3]], [4, 1, 1, 1, 1], [3, 1, 1, 1, 0], [-1, 0, 0, 0, -1], -2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
70. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[1, 2]], [0, 4], added, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2]], [0, 4], added, [[0, 4], [1, 2]], [0, 1, 1, 0, 0], [1, 1, 1, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
71. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [1, 2]], [0, 4], added, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 4], added, [[0, 1], [0, 4], [1, 2]], [1, 2, 1, 0, 0], [2, 2, 1, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
72. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 2], [1, 2]], [0, 4], added, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 2]], [0, 4], added, [[0, 2], [0, 4], [1, 2]], [1, 1, 2, 0, 0], [2, 1, 2, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
73. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 2], [1, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [1, 2]], [0, 4], added, [[0, 1], [0, 2], [0, 4], [1, 2]], [2, 2, 2, 0, 0], [3, 2, 2, 0, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
74. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 3], [1, 2]], [0, 4], added, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [1, 2]], [0, 4], added, [[0, 3], [0, 4], [1, 2]], [1, 1, 1, 1, 0], [2, 1, 1, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
75. Toggle a Typed Nonloop Edge and Compute Degree Changes
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).
View answer and explanation
([[0, 1], [0, 3], [1, 2]], [0, 4], added, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2)
Remove an existing edge or insert an absent one, then independently count both degree lists; no loop or invalid endpoint is admitted. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [1, 2]], [0, 4], added, [[0, 1], [0, 3], [0, 4], [1, 2]], [2, 2, 1, 1, 0], [3, 2, 1, 1, 1], [1, 0, 0, 0, 1], 2).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-edge-toggle-delta · candidate
76. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([], [], [], [], [])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], [], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
77. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [1, 2]], [], [], [], [])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [], [], [], []).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
78. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[0, 2], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2], [], [0, 0], [[], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
79. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[2, 1], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [1, 2], [[1, 2]], [1, 1], [[2], [1]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
80. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[3, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 2, 3], [[2, 3]], [0, 1, 1], [[], [3], [2]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
81. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
82. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
83. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
84. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
85. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
86. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
87. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
88. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
89. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
90. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
91. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
92. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
93. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4]], [0, 2, 4], [[0, 4]], [1, 0, 1], [[4], [], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
94. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
95. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
96. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
97. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
98. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
99. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [1, 2]], [0, 2, 4], [[0, 2]], [1, 1, 0], [[2], [0], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
100. Induced Graph with Original Vertex Labels
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).
View answer and explanation
([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []])
Keep exactly edges with both endpoints retained, and do not relabel the induced vertices. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [1, 2]], [0, 2, 4], [], [0, 0, 0], [[], [], []]).
Reviewed step-by-step derivation
Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate
Canonical question JSONL · English JSONL · Validation report