Question 8089: Induced Graph with Original Vertex Labels

Original question and givens

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).

• G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).

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, 1], [0, 4]]

canonical edges=[[0, 1], [0, 4]]

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

neighbors=[1, 4];degree=2

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

neighbors=[0];degree=1

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

neighbors=[];degree=0

phase=original;vertex=3;all edges=[[0, 1], [0, 4]]

neighbors=[];degree=0

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

neighbors=[0];degree=1

raw subset=[4, 2, 0];old edges=[[0, 1], [0, 4]]

retained original labels=[0, 2, 4];induced edges=[[0, 4]]

phase=induced;vertex=0;all edges=[[0, 4]]

neighbors=[4];degree=1

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

neighbors=[];degree=0

phase=induced;vertex=4;all edges=[[0, 4]]

neighbors=[0];degree=1

Reverse checks

• 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

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

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.