# Question 8060: Toggle a Typed Nonloop Edge and Compute Degree Changes [Return to catalogue page 41](https://ocme.evemisslab.com/en/questions/page/41/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r05-graph-foundations/#q-20261012-advance-20261017-r05-graph-0060) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0015-08060 · q-20261012-advance-20261017-r05-graph-0060 ## Original question and givens Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[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. 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]]. 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](https://ocme.evemisslab.com/en/mko/mko-graph/) ## Step-by-step derivation S001 · impliesBefore n=5;raw edges=[[0, 2], [0, 3]] After canonical edges=[[0, 2], [0, 3]] Conditions: none Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing. Step identifiers and dependenciesRule: simple_graph_edge_normalization · Uses: G001 S002 · impliesBefore phase=original;vertex=0;all edges=[[0, 2], [0, 3]] After neighbors=[2, 3];degree=2 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S001 S003 · impliesBefore phase=original;vertex=1;all edges=[[0, 2], [0, 3]] After neighbors=[];degree=0 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S002 S004 · impliesBefore phase=original;vertex=2;all edges=[[0, 2], [0, 3]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S003 S005 · impliesBefore phase=original;vertex=3;all edges=[[0, 2], [0, 3]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S004 S006 · impliesBefore phase=original;vertex=4;all edges=[[0, 2], [0, 3]] After neighbors=[];degree=0 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S005 S007 · impliesBefore typed candidate=[0, 4];old edges=[[0, 2], [0, 3]] After action=added;new edges=[[0, 2], [0, 3], [0, 4]] Conditions: none Normalize a reversed candidate first,then decide addition/removal from the original edge set.Exactly one undirected edge changes. Step identifiers and dependenciesRule: toggle_one_undirected_nonloop_edge · Uses: G001, S006 S008 · impliesBefore phase=toggled;vertex=0;all edges=[[0, 2], [0, 3], [0, 4]] After neighbors=[2, 3, 4];degree=3 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S007 S009 · impliesBefore phase=toggled;vertex=1;all edges=[[0, 2], [0, 3], [0, 4]] After neighbors=[];degree=0 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S008 S010 · impliesBefore phase=toggled;vertex=2;all edges=[[0, 2], [0, 3], [0, 4]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S009 S011 · impliesBefore phase=toggled;vertex=3;all edges=[[0, 2], [0, 3], [0, 4]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S010 S012 · impliesBefore phase=toggled;vertex=4;all edges=[[0, 2], [0, 3], [0, 4]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S011 S013 · calculationBefore degree delta0 := sub(3, 2) After degree delta0 = 1 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore degree delta1 := sub(0, 0) After degree delta1 = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore degree delta2 := sub(1, 1) After degree delta2 = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore degree delta3 := sub(1, 1) After degree delta3 = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore degree delta4 := sub(1, 0) After degree delta4 = 1 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore degree delta sum := sum(1, 0, 0, 0, 1) After degree delta sum = 2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 ## 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], added, [[0, 2], [0, 3], [0, 4]], [2, 0, 1, 1, 0], [3, 0, 1, 1, 1], [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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 271c5bd76364e51c94bbde720cbbda52aeea49baf3963432c9b6db852ed9c283 - candidate record SHA-256: 35d3981fd08b0acad9f2c84dd070ad0054dc3759719ebb9ebc3139ff3fff2b9f - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r05-graph-0060.json)