Nonempty Tree Qualification, Paths, Leaves, and New-Edge Cycles
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261012-advance-20261017-r06-tree-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. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(0, [], [], Yes, No, No, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], Yes, No, No, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
2. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(1, [], [[0]], Yes, No, Yes, Yes)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[0]], Yes, No, Yes, Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
3. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], No, Yes, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0], [1, 2, 3]], No, Yes, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
4. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
5. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 1, 2, 3]], Yes, No, Yes, Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
6. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 1, 2, 3]], Yes, No, Yes, Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
7. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
8. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
9. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
10. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
11. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
12. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
13. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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, 1, 2, 3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
14. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
15. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
16. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
17. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 1, 2, 4], [3]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
18. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
19. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 1, 3, 4], [2]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
20. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 2, 3, 4], [1]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
21. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes)
Check the full component and cycle structure, not just a numerical edge condition. 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]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
22. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
23. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
24. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
25. Components, Cycles, and Nonempty Tree Qualification
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
View answer and explanation
(5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, Yes, Yes, No)
Check the full component and cycle structure, not just a numerical edge condition. 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, 1, 2], [3], [4]], No, Yes, Yes, No).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate
26. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=-1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(0, [], No, [No, No], not-a-tree, not-evaluated)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], not-a-tree, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
27. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(1, [], Yes, [Yes, Yes], [0], 0)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], [0], 0).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
28. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], not-a-tree, not-evaluated)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. 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, [Yes, Yes], not-a-tree, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
29. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=-1, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
30. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
31. Original-Tree and Endpoint Guards before a Unique Path
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. 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]], Yes, [Yes, Yes], [1, 0, 3], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
32. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
33. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
34. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
35. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
36. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
37. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
38. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
39. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
40. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
41. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
42. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
43. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
44. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
45. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
46. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
47. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
48. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
49. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
50. Original-Tree and Endpoint Guards before a Unique Path
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
View answer and explanation
(6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1)
Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate
51. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
52. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
53. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
54. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], No, Yes, 0, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. 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, Yes, 0, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
55. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
56. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=3. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
57. Degree-One Leaf Deletion without Renumbering
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=2. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(4, [[0, 1], [0, 2], [0, 3]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. 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]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
58. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
59. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
60. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
61. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
62. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
63. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
64. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
65. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
66. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
67. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
68. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
69. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
70. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
71. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
72. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
73. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
74. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
75. Degree-One Leaf Deletion without Renumbering
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)
A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate
76. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
77. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
78. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
79. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[2, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
80. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[-1, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
81. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[1, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
82. Qualified Tree Nonedge and Its Unique Cycle Witness
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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. 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, [Yes, Yes], Yes, No, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
83. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
84. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
85. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
86. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
87. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
88. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
89. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
90. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
91. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
92. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
93. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
94. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
95. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
96. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
97. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
98. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
99. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
100. Qualified Tree Nonedge and Its Unique Cycle Witness
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [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. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.
View answer and explanation
(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])
Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).
Reviewed step-by-step derivation
Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate
Canonical question JSONL · English JSONL · Validation report