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.