Path partitions and image-free sets

The detour order image of a graph G is the order of a longest path of G. If S is a subset of image such that the graph induced by S has detour order at most n, then S is called a image-free set in G. The Path Partition Conjecture (PPC) can be stated as follows: For any graph G and any positive integer image there exists a image-free set H in G such that image. We prove that if G is any graph and M is any maximal image-free set in G, then image. We also prove that if G has no cycle of order less than n or greater than image, then image for every maximal image-free subset M of image. As a corollary of the latter result we prove that the PPC is true for the class of connected, weakly pancyclic graphs.