Resolvability and Strong Resolvability in the Direct Product of Graphs

TL;DR

This paper investigates the properties of resolvability and strong resolvability in the direct product of graphs, focusing on how these concepts influence the metric and strong metric dimensions of such graph families.

Contribution

It introduces new results on the metric and strong metric dimensions specifically for direct product graphs, expanding understanding of their structural properties.

Findings

Determined the metric dimension for certain classes of direct product graphs.

Established bounds for the strong metric dimension in these graph families.

Provided characterizations of resolving sets in the context of direct product 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.