The search session has expired. Please query the service again.
The search session has expired. Please query the service again.
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...
We prove that for every number , the -iterated -path graph of is isomorphic to if and only if is a collection of cycles, each of length at least 4. Hence, is isomorphic to if and only if is a collection of cycles, each of length at least 4. Moreover, for we reduce the problem of characterizing graphs such that to graphs without cycles of length exceeding .
Download Results (CSV)