Histories in path graphs
For a given graph G and a positive integer r the r-path graph, , 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 be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of . The k-history is a subgraph of G that is induced by all edges that take part in the recursive definition of...