The local metric dimension of strong product graphs

TL;DR

This paper investigates the local metric dimension of strong product graphs, providing exact values or bounds, which is significant given the NP-Completeness of the problem.

Contribution

It offers new results on calculating or bounding the local metric dimension specifically for strong product graphs.

Findings

Derived bounds for local metric dimension of strong product graphs

Identified exact values for specific classes of strong product graphs

Enhanced understanding of the complexity related to local metric dimension

Abstract

A vertex $v\in V(G)$ is said to distinguish two vertices $x,y\in V(G)$ of a nontrivial connected graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subset V(G)$ is a local metric generator for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex of $S$. A local metric generator with the minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of $G$. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.