Summary: We prove that for every number \(n\geq 1\), 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 \(k\geq 4\) we reduce the problem of characterizing graphs \(G\) such that \(P_k(G)\cong G\) to graphs without cycles of length exceeding \(k\).