On the strong metric dimension of Cartesian sum graphs

TL;DR

This paper investigates the strong metric dimension of Cartesian sum graphs, providing bounds and formulas based on properties of the factor graphs such as clique numbers and twins-free clique numbers.

Contribution

It introduces new bounds and closed-form expressions for the strong metric dimension of Cartesian sum graphs using properties of the factor graphs.

Findings

Derived tight bounds for strong metric dimension

Established formulas involving clique and twins-free clique numbers

Applied results to specific classes of graphs

Abstract

A vertex $w$ of a connected graph $G$ 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 tight bounds or closed formulae for the strong metric dimension of the Cartesian sum of graphs in terms of the strong metric dimension, clique number or twins-free clique number of its factor graphs.