Histories in path graphs

For a given graph G and a positive integer r the r-path graph, ${\displaystyle P_r\left(G\right)}$, 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 ${\displaystyle P^k\_r\left(G\right)}$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of ${\displaystyle P^k\_r\left(G\right)}$. The k-history ${\displaystyle P^{-k}\_r\left(H\right)}$ 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.