A sufficient condition for Pk-path graphs being r-connected

Abstract

Given an integer k≥1 and any graph G; the path graph $P_k(G)$ has for vertices the paths of length k in G, and two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k-1 in G, and their union forms either a cycle or a path of length k + 1. Path graphs were investigated by Broersma and Hoede [Path graphs, J. Graph Theory 13 (1989), 427-444] as a natural generalization of line graphs. In fact, $P_1(G)$ is the line graph of G: For k = 1,2 results on connectivity of $P_k(G)$ have been given for several authors. In this work we present a sufficient condition to guarantee that $P_k(G)$ is connected for k≥2 if the girth of G is at least (k+3)/2 and its minimum degree is at least 4. Furthermore, we determine a lower bound of the vertex-connectivity of $P_k(G)$ if the girth is at least k+1 and the minimum degree is at least r + 1 where r ≥ 2 is an integer.