# Simple Undirected Graph Inventories and Induced Subsets

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r05-graph-0100/)

Finite simple undirected graphs · qf-density17-graph-induced-subset · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r05-graph-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261017-r05-graph-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r05-graph-foundations/validation-report.json)