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Question 13689: Reflexive Connectivity Includes a Length-Zero Path

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0026-13689 · q-20261014-advance-20261028-r07-connected-component-0064

Original question and givens

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).

  • G001 — Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).

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.

Step-by-step derivation

S001 · implies

Before

full carrier Fin(7);raw undirected edges=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]

After

canonical simple edges=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]];full adjacency rows=[[2, 5], [4], [0, 4, 5], [4, 5], [1, 2, 3], [0, 2, 3], []]

Conditions: none

Build adjacency on the full carrier,rejecting loops and duplicate undirected edges.Vertices absent from edges retain empty rows;do not switch to directed connectivity.

Step identifiers and dependencies

Rule: component_full_undirected_graph_includes_isolates · Uses: G001

S002 · implies

Before

left vertex:start vertex=6;complete adjacency=[[2, 5], [4], [0, 4, 5], [4, 5], [1, 2, 3], [0, 2, 3], []]

After

initial visited=[6];path to start=[6];edge length0

Conditions: none

The start reaches itself by a length-zero path.Even an isolated vertex must be included rather than omitted for having no edges.

Step identifiers and dependencies

Rule: component_zero_length_reflexive_path · Uses: G001, S001

S003 · implies

Before

left vertex:layer=0;frontier=[6];all frontier neighbors=[]

After

previously seen=[6];new vertices=[]

Conditions: none

At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment.

Step identifiers and dependencies

Rule: component_complete_frontier_expansion · Uses: G001, S002

S004 · implies

Before

left vertex:final visited=[6];all [vertex,neighbors,outside neighbors] rows=[[6, [], []]]

After

no edge leaves visited;every visited vertex has a path;visited equals full reachable set

Conditions: none

Path witnesses give visited⊆reachable.At completion no edge leaves visited.If a path reached outside,it would have a first leaving edge,a contradiction,so reachable⊆visited.Both directions establish the whole component,not a traversal order.

Step identifiers and dependencies

Rule: component_reachable_and_closed_set_completeness · Uses: G001, S003

S005 · implies

Before

left=6;right=5;whole left component=[6]

After

same component=No

Conditions: none

Compute the whole left component before testing the right label.Equal endpoints are connected by a zero-length path even if isolated.

Step identifiers and dependencies

Rule: component_membership_after_complete_reachability · Uses: G001, S004

Reverse checks

  • 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

(6, 5, [6], 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: 150a38ca953fa3c458c11199ed102e3c745c309e9e65307076ffb4d469338def
  • candidate record SHA-256: 255136f4c6eeb44cfff53ad5c494c9b7124280218d0e018b9767223d30292130
  • semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798
  • question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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