Fig. 2
From: Non-backtracking cycles: length spectrum theory and graph mining applications
Conjugate homotopy classes in a graph. Left: In an undirected graph G we label some directed edges a,b,c,d and e. We use the red node as basepoint to construct homotopy classes. Center: Three homotopy classes [h1],[h2],[g] each with a sequence of edges that form a cycle of minimal length. Right: Proof that [h1] and [h2] are conjugate via [g]. Each time an edge is followed by its inverse we remove them as they are homotopically trivial. By doing so, we reduce the cycle [g−1][h1][g] to one that is homotopic to it and has minimal-length, namely [h2]