DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite simplicial paths, reduced paths, cycles, and connectedness

Definition

Let $X = (V, E)$ be an oriented graph in the sense of An oriented graph with edge reversal.

A path in $X$ is a finite alternating sequence

$v_{0}, e_{1}, v_{1}, \dots, e_{n}, v_{n}$

with $o (e_{i}) = v_{i - 1}$ and $t (e_{i}) = v_{i}$ for each $i$ . Its length is $n$ . The path is closed when $v_{0} = v_{n}$ , and reduced when

$e_{i + 1} \neq {\overset{ˉ}{e}}_{i} (1 \leq i < n) .$

A cycle is a nontrivial reduced closed path. Vertices $v, w \in V$ are connected when some path joins them, and $X$ is connected when every two vertices are connected.

All paths on this page are finite, even when $V$ is infinite.

Depends on

An oriented graph with edge reversal

Used by

A simplicial tree Definition

Dependency tree · one level

1 result within one dependency step 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

Jean-Pierre Serre, Trees (standard reference, not scraped)
Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)