Closed formulae for the strong metric dimension of lexicographic product graphs

TL;DR

This paper derives explicit formulas for calculating the strong metric dimension of lexicographic product graphs based on the properties of their factor graphs, advancing understanding of graph metric parameters.

Contribution

It provides new closed-form expressions linking the strong metric dimension of lexicographic products to that of individual factor graphs.

Findings

Derived relationships between strong metric dimensions of product and factor graphs.

Established formulas for the strong metric dimension of lexicographic product graphs.

Enhanced methods for calculating graph metric parameters in complex networks.

Abstract

Given a connected graph $G$, 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 $G$. In this paper we obtain several relationships between the strong metric dimension of the lexicographic product of graphs and the strong metric dimension of its factor graphs.