LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Being joined by a walk or path is the same equivalence relation on the vertex set

Statement

On the vertex set of a graph, define $u \sim v$ when a walk joins $u$ to $v$ . Then $\sim$ is an equivalence relation (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ), and $u \sim v$ holds exactly when a path joins $u$ to $v$ .

Facts & Assumptions

Given: A graph $G = (V, E)$ and the relation $u \sim v$ defined by the existence of a walk from $u$ to $v$ .

[F1]

An equivalence relation is reflexive, symmetric and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

[L1]

Every walk between two vertices contains a path between the same endpoints (Every walk between two vertices contains a path between the same endpoints).

[F2]

A one-vertex list is a path of length $0$ , reversing a walk gives a walk with reversed endpoints, and concatenating a $u$ to $v$ walk with a $v$ to $w$ walk gives a $u$ to $w$ walk (Connected graphs and connected components defined by the existence of vertex paths, Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

Proof

technique · direct

1.1

For every $u \in V$ , the length- $0$ walk $(u)$ joins $u$ to itself, so $u \sim u$ .

1.2

If $u \sim v$ , reversing a witnessing walk gives a walk from $v$ to $u$ , so $v \sim u$ .

givenF2

1.3

If $u \sim v$ and $v \sim w$ , concatenating witnessing walks gives a walk from $u$ to $w$ , so $u \sim w$ .

givenF2

2.1

Steps 1.1, 1.2 and 1.3 prove reflexivity, symmetry and transitivity, hence $\sim$ is an equivalence relation by [F1].

step 1.1step 1.2step 1.3F1

3.1

Every path is a walk, while [L1] turns every witnessing walk into a path with the same endpoints; therefore walk-reachability and path-reachability coincide.

L1F2∎

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Lethbridge, Combinatorics: A Guided Tour, Paths and Cycles (standard reference, not scraped)