Resolvability and strong resolvability in the direct product of graphs

Abstract

Given a connected graph $G$, a vertex $w\in V(G)$ distinguishes two different vertices $u,v$ of $G$ if the distances between $w$ and $u$, and between $w$ and $v$ are different. Moreover, $w$ strongly resolves the pair $u,v$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a (strong) metric generator for $G$ if every pair of vertices of $G$ is (strongly resolved) distinguished by some vertex of $W$. The smallest cardinality of a (strong) metric generator for $G$ is called the (strong) metric dimension of $G$. In this article we study the (strong) metric dimension of some families of direct product graphs.