On the strong metric generators of strong product graphs

TL;DR

This paper investigates the strong metric dimension of strong product graphs, providing exact values and bounds expressed through the properties of the factor graphs, addressing an NP-hard problem.

Contribution

It offers new bounds and exact formulas for the strong metric dimension of strong product graphs based on their factor graphs' invariants.

Findings

Derived bounds for strong metric dimension of strong product graphs

Provided exact values for specific classes of strong product graphs

Connected the strong metric dimension to invariants of factor graphs

Abstract

Let $G$ be a connected graph. A vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.