Walks, Paths and Cycles

Abstract

A walk in a graph G is a finite sequence of vertices x ₀, x ₁, ..., x _n and edges a ₁, a ₂, ..., a _n of G:

$${x_0},{a_1},{x_1},{a_2}, \ldots ,{a_n},{x_n},$$

where the endpoints of a _i are x _i−1 and x _i for each i. A simple walk is a walk in which no edge is repeated. A path is a walk in which no vertex is repeated; the length of a path is its number of edges. A walk is closed when the first and last vertices, x ₀ and x _n, are equal. A cycle of length n is a closed simple walk of length n, n ≥ 3, in which the vertices x ₀, x ₁, ..., x _n−1 are all different.