Edge-connectivity in Pk-path graphs

Abstract

The P k (G)-path graph corresponding to a graph G has for vertices the set of all paths of length k in G. 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 introduced by Broersma and Hoede (J. Graph. Theory 13 (1989) 427) as a generalization of line graphs, because for k = 1, path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (12). The connectivity of P k -path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the P k (G)-path graphs to be disconnected, assuming that G has girth of at least k + 1. Going one step further, we prove in this work that the edge-connectivity of P k (G) is at least (P k (G)) (G) − 1 for a graph G of girth at least k + 1 and minimum degree (G) 2. Furthermore, we show (P k (G)) 2 (G) − 2 provided that (G) 3.