On the strong metric dimension of Cartesian and direct products of graphs

TL;DR

This paper derives explicit formulas for the strong metric dimension of various Cartesian and direct product graphs, addressing an NP-hard problem in graph theory.

Contribution

It provides the first closed-form expressions for the strong metric dimension of several classes of product graphs.

Findings

Closed formulas for Cartesian product graphs.

Closed formulas for direct product graphs.

Addresses NP-hardness in computing strong metric dimension.

Abstract

Let $G$ be a connected graph. A vertex $w$ {\em strongly resolves} a pair $u, v$ of vertices of $G$ 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 {\em strong resolving set} for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. The smallest cardinality of a strong resolving set for $G$ is called the {\em strong metric dimension} of $G$. It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian product graphs and direct product graphs.