Path

Definition

Let $G$ be a graph. A path in $G$ is a walk $(w_0, w_1, \dots, w_k)$ whose vertices $w_0, w_1, \dots, w_k$ are pairwise distinct.

Remarks

A path uses each edge at most once, so every path is a trail. A path of length $k$ has $k + 1$ vertices. Its vertices and edges form a subgraph isomorphic to the path graph $P_{k+1}$. Each walk from $u$ to $v$ contains a path from $u$ to $v$. The node every walk contains a path states this.

Explanations

L1 human

A path is a walk that never visits the same vertex twice. On a map, it is a route that never passes through the same crossing again.