Strong metric dimension of rooted product graphs

TL;DR

This paper investigates the strong metric dimension of rooted product graphs, providing exact values and bounds expressed through the properties of the component graphs, addressing the computational complexity of the problem.

Contribution

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

Findings

Exact values for the strong metric dimension of rooted product graphs.

Sharp bounds expressed in terms of factor graph invariants.

Addresses NP-hardness by focusing on special graph classes.

Abstract

Let $G$ be a connected graph. A vertex $w$ 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 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 strong metric dimension of $G$. It is known that the problem of computing this invariant is NP-hard. This suggests finding the strong metric dimension for special classes of graphs or obtaining good bounds on this invariant. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of rooted product of graphs and express these in terms of invariants of the factor graphs.