Abstract. If G is a graph, then its path graph, P_k(G), has vertex set identical with the set of paths of length k in G, with two vertices adjacent in P_k(G) if and only if the corresponding paths are "consecutive" in G. Observe that P₁(G) is the line graph of G. It is known that if k<3, then a connected graph G is isomorphic to its path graph P_k(G) if and only if G is a cycle. We prove here that a connected graph G is isomorphic to its path graph P₃(G) if and only if G is a cycle of length at least 4. In fact, we prove more. We prove that i-iterated P₃-path graph of G is isomorphic to G if and only if G is a collection of isolated cycles, each of length at least 4. Moreover, for k>3 we reduce the problem of characterizing graphs G, such that P_k(G) is isomorphic to G, to graphs without cycles of length exceeding k.