Graphs isomorphic to their path graphs

Abstract

We prove that for every number $n\ge 1$, the $n$-iterated $P_3$-path graph of $G$ is isomorphic to $G$ if and only if $G$ is a collection of cycles, each of length at least 4. Hence, $G$ is isomorphic to $P_3(G)$ if and only if $G$ is a collection of cycles, each of length at least 4. Moreover, for $k\ge 4$ we reduce the problem of characterizing graphs $G$ such that $P_k(G)\cong G$ to graphs without cycles of length exceeding $k$.