Containers and wide diameters of $P_3(G)$

Abstract

The P 3 intersection graph of a graph G has for vertices all the induced paths of order 3 in G. Two vertices in P 3 (G) are adjacent if the corresponding paths in G are not disjoint. A w-container between two different vertices u and v in a graph G is a set of w internally vertex disjoint paths between u and v. The length of a container is the length of the longest path in it. The w-wide diameter of G is the minimum number l such that there is a w-container of length at most l between any pair of different vertices u and v in G. Interconnection networks are usually modeled by graphs. The w-wide diameter provides a measure of the maximum communication delay between any two nodes when up to w − 1 nodes fail. Therefore, the wide diameter constitutes a measure of network fault tolerance. In this paper we construct containers in P 3 (G) and apply the results obtained to the study of their connectivity and wide diameters.