Walk

Definition

Let $G$ be a graph, let $k$ be a natural number, and let $u, v$ be vertices of $G$. A walk of length $k$ from $u$ to $v$ is a sequence of vertices $(w_0, w_1, \dots, w_k)$ with $w_0 = u$, $w_k = v$, and $w_{i-1} \sim w_i$ for $1 \le i \le k$. Its edges are the edges $w_{i-1} w_i$.

Remarks

A walk may repeat vertices and edges. A walk of length $0$ is a single vertex. A trail has no repeated edge, and a path has no repeated vertex. A walk is closed if $u = v$.

Explanations

L1 human

A walk is a route through a graph that goes from vertex to vertex along edges. You may visit the same vertex or use the same edge more than once.