# Question 13368: Distinguish Repeated Vertices, Repeated Edges and Closed Walks [Return to catalogue page 67](https://ocme.evemisslab.com/en/questions/page/67/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations/#q-20261014-advance-20261027-r07-graph-walk-simple-path-0093) 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. R0025-13368 · q-20261014-advance-20261027-r07-graph-walk-simple-path-0093 ## Original question and givens Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate. - G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate. ## 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-walk-simple-path](https://ocme.evemisslab.com/en/mko/mko-graph-walk-simple-path/) ## Step-by-step derivation S001 · impliesBefore carrier Fin(6);raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];original vertex sequence=[2, 4, 0, 1, 0] After canonical undirected simple edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];nonempty typed sequence retained Conditions: none Normalize undirected graph edges and reject loops/reversed duplicates.Preserve vertex-sequence order and repetitions.Repetition is not a parsing error in a candidate,though it may fail walk or simplicity qualification. Step identifiers and dependenciesRule: walk_simple_graph_and_ordered_candidate · Uses: G001 S002 · impliesBefore proposed step index=0;from=2;to=4;unordered step edge=[2, 4] After edge present=Yes Conditions: none Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity. Step identifiers and dependenciesRule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S001 S003 · impliesBefore proposed step index=1;from=4;to=0;unordered step edge=[0, 4] After edge present=Yes Conditions: none Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity. Step identifiers and dependenciesRule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S002 S004 · impliesBefore proposed step index=2;from=0;to=1;unordered step edge=[0, 1] After edge present=Yes Conditions: none Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity. Step identifiers and dependenciesRule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S003 S005 · impliesBefore proposed step index=3;from=1;to=0;unordered step edge=[0, 1] After edge present=Yes Conditions: none Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity. Step identifiers and dependenciesRule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S004 S006 · impliesBefore all bad proposed steps=[] After walk=Yes Conditions: none The whole step list must pass.A singleton has no proposed steps and passes vacuously as a valid length-zero walk. Step identifiers and dependenciesRule: complete_walk_candidate_qualification · Uses: G001, S005 S007 · calculationBefore proposed edge length := sub(5, 1) After proposed edge length = 4 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore original vertex sequence=[2, 4, 0, 1, 0] After sorted distinct repeated vertex labels=[0] Conditions: none A simple path forbids any repeated vertex,not only immediate repeats.This includes a closed walk's repeated endpoint. Step identifiers and dependenciesRule: simple_path_vertex_repetition_full_sequence · Uses: G001, S007 S009 · impliesBefore proposed step index=0;unordered edge=[2, 4] After seen at earlier step=No;distinct repeated edges so far=[] Conditions: none Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order. Step identifiers and dependenciesRule: trail_undirected_edge_first_repeat_order · Uses: G001, S008 S010 · impliesBefore proposed step index=1;unordered edge=[0, 4] After seen at earlier step=No;distinct repeated edges so far=[] Conditions: none Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order. Step identifiers and dependenciesRule: trail_undirected_edge_first_repeat_order · Uses: G001, S009 S011 · impliesBefore proposed step index=2;unordered edge=[0, 1] After seen at earlier step=No;distinct repeated edges so far=[] Conditions: none Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order. Step identifiers and dependenciesRule: trail_undirected_edge_first_repeat_order · Uses: G001, S010 S012 · impliesBefore proposed step index=3;unordered edge=[0, 1] After seen at earlier step=Yes;distinct repeated edges so far=[[0, 1]] Conditions: none Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order. Step identifiers and dependenciesRule: trail_undirected_edge_first_repeat_order · Uses: G001, S011 S013 · impliesBefore walk=Yes;length=4;start=2;end=0;repeated vertices=[0];repeated edges=[[0, 1]] After trail=No;simple path=No;closed positive-length walk=No Conditions: none A trail has no repeated edges,a simple path no repeated vertices,and closed additionally requires positive length and equal endpoints.These are distinct.A singleton is a trail/simple path but is not closed in this protocol;closed is not a simple-cycle certificate. Step identifiers and dependenciesRule: walk_trail_simple_closed_separate_obligations · Uses: G001, S012 ## Reverse checks - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute proposed edge length 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 (Yes, 4, [0], [[0, 1]], No, No, No) 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: ecdbccfcb96e5168fb2b3c78d65f54c3b41d54d1c3fa8978adb88a144da151df - candidate record SHA-256: 56fac5038ffca1d98368cb91f5b0387f955810655dad3d671a69b0892e6b0ba5 - semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5 - question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0093.json)