Question 8481: Original-Tree and Endpoint Guards before a Unique Path

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-tree

Step-by-step derivation

n=4;raw edges=[[0, 1], [0, 2], [0, 3]]

canonical edges=[[0, 1], [0, 2], [0, 3]]

phase=original;vertex=0;all edges=[[0, 1], [0, 2], [0, 3]]

neighbors=[1, 2, 3];degree=3

phase=original;vertex=1;all edges=[[0, 1], [0, 2], [0, 3]]

neighbors=[0];degree=1

phase=original;vertex=2;all edges=[[0, 1], [0, 2], [0, 3]]

neighbors=[0];degree=1

phase=original;vertex=3;all edges=[[0, 1], [0, 2], [0, 3]]

neighbors=[0];degree=1

phase=original;root=0;depth=0;frontier=[0]

next newly reached=[1, 2, 3];component seen=[0, 1, 2, 3]

phase=original;root=0;depth=1;frontier=[1, 2, 3]

next newly reached=[];component seen=[0, 1, 2, 3]

phase=original;spanning forest=[[0, 1], [0, 2], [0, 3]];all edges=[[0, 1], [0, 2], [0, 3]]

nonforest edges=[];cyclic=No

phase=original;vertices=[0, 1, 2, 3];components=[[0, 1, 2, 3]];cyclic=No

pairwise reachable=Yes;nonempty=Yes;tree=Yes

original tree=Yes;endpoints=[1, 3];typed=[Yes, Yes]

path lookup qualified=Yes

phase=query;prefix=[1];next=0

edge=[0, 1] present;new prefix=[1, 0]

phase=query;prefix=[1, 0];next=3

edge=[0, 3] present;new prefix=[1, 0, 3]

phase=query;start=1;end=3

unique path=[1, 0, 3]

qualified path edge count := sub(3, 1)

qualified path edge count = 2

Reverse checks

• S014 · premise_and_result_audit — Recompute qualified path edge count with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.