Question 8068: Toggle a Typed Nonloop Edge and Compute Degree Changes

Original question and givens

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).

• G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Toggle edge=[0, 4]. Give (original normalized edges, normalized toggled edge, removed/added action, new edges, old degrees, new degrees, degree differences, degree-sum difference).

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], [0, 3], [0, 4]]

canonical edges=[[0, 2], [0, 3], [0, 4]]

phase=original;vertex=0;all edges=[[0, 2], [0, 3], [0, 4]]

neighbors=[2, 3, 4];degree=3

phase=original;vertex=1;all edges=[[0, 2], [0, 3], [0, 4]]

neighbors=[];degree=0

phase=original;vertex=2;all edges=[[0, 2], [0, 3], [0, 4]]

neighbors=[0];degree=1

phase=original;vertex=3;all edges=[[0, 2], [0, 3], [0, 4]]

neighbors=[0];degree=1

phase=original;vertex=4;all edges=[[0, 2], [0, 3], [0, 4]]

neighbors=[0];degree=1

typed candidate=[0, 4];old edges=[[0, 2], [0, 3], [0, 4]]

action=removed;new edges=[[0, 2], [0, 3]]

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

neighbors=[2, 3];degree=2

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

neighbors=[];degree=0

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

neighbors=[0];degree=1

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

neighbors=[0];degree=1

phase=toggled;vertex=4;all edges=[[0, 2], [0, 3]]

neighbors=[];degree=0

degree delta0 := sub(2, 3)

degree delta0 = -1

degree delta1 := sub(0, 0)

degree delta1 = 0

degree delta2 := sub(1, 1)

degree delta2 = 0

degree delta3 := sub(1, 1)

degree delta3 = 0

degree delta4 := sub(0, 1)

degree delta4 = -1

degree delta sum := sum(-1, 0, 0, 0, -1)

degree delta sum = -2

Reverse checks

• S018 · premise_and_result_audit — Recompute degree delta sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute degree delta4 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute degree delta3 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute degree delta2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute degree delta1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute degree delta0 with exact rational arithmetic, not rounded decimals.
• 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

([[0, 2], [0, 3], [0, 4]], [0, 4], removed, [[0, 2], [0, 3]], [3, 0, 1, 1, 1], [2, 0, 1, 1, 0], [-1, 0, 0, 0, -1], -2)

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.