Question 8052: Toggle a Typed Nonloop Edge and Compute Degree Changes

Original question and givens

Use a complete simple undirected graph on Fin(2)={0,…,1} 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. Toggle edge=[0, 1]. 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(2)={0,…,1} 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. Toggle edge=[0, 1]. 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=2;raw edges=[]

canonical edges=[]

phase=original;vertex=0;all edges=[]

neighbors=[];degree=0

phase=original;vertex=1;all edges=[]

neighbors=[];degree=0

typed candidate=[0, 1];old edges=[]

action=added;new edges=[[0, 1]]

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

neighbors=[1];degree=1

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

neighbors=[0];degree=1

degree delta0 := sub(1, 0)

degree delta0 = 1

degree delta1 := sub(1, 0)

degree delta1 = 1

degree delta sum := sum(1, 1)

degree delta sum = 2

Reverse checks

• S009 · premise_and_result_audit — Recompute degree delta sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute degree delta1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute degree delta0 with exact rational arithmetic, not rounded decimals.
• 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], added, [[0, 1]], [0, 0], [1, 1], [1, 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.