Histories in Path Graphs

For a given graph G and a positive integer r the r-path graph, Pr(G), has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r − 1, and their union forms either a cycle or a path of length k+1 in G. Let P k r (G) be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of P k r (G). The k-history P r (H) is a subgraph of G that is induced by all edges that take part in the recursive definition of H . We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.