Abstract. If G is a graph, then its P_k-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. We prove a necessary and sufficient condition under which a connected graph has a connected P₃-path graph. Moreover, an analogous condition for connectivity of the P_k-path graph which does not contain a cycle of length smaller than k+1 is derived.