Question 8049: Every Vertex Neighbor Partition and Isolated Vertices

Original question and givens

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).

• G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).

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-graph

Step-by-step derivation

n=5;raw edges=[[0, 2], [1, 2]]

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

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

neighbors=[2];degree=1

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

neighbors=[2];degree=1

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

neighbors=[0, 1];degree=2

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

neighbors=[];degree=0

phase=original;vertex=4;all edges=[[0, 2], [1, 2]]

neighbors=[];degree=0

vertices=[0, 1, 2, 3, 4];neighbors=[[2], [2], [0, 1], [], []]

nonneighbors excluding self=[[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]];isolated=[3, 4]

Reverse checks

• 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

(5, [[0, 2], [1, 2]], [[2], [2], [0, 1], [], []], [[1, 3, 4], [0, 3, 4], [3, 4], [0, 1, 2, 4], [0, 1, 2, 3]], [3, 4])

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.