On the Metric Dimension of Generalized Petersen Graphs $P(n,3)$

TL;DR

This paper determines the metric dimension of generalized Petersen graphs P(n,3), proving it equals 4 for large n with specific modular conditions, using notions of good and bad vertices to establish bounds.

Contribution

It introduces an approach based on good and bad vertices to determine the metric dimension of P(n,3), establishing exact values for large n under certain conditions.

Findings

Metric dimension of P(n,3) is 4 for large n with n mod 6 in {2,3,4,5}.

Introduces a method using good and bad vertices to find lower bounds.

Provides exact metric dimension values for specific classes of generalized Petersen graphs.

Abstract

The metric dimension of a graph $G$ is defined as the minimum number of vertices in a subset $S\subset V(G)$ such that all other vertices are uniquely determined by their distances to the vertices in $S$, and is denoted by $\dim(G)$. In this paper, we study the metric dimension of generalized Petersen graphs $P(n,3)$. The notions of good and bad vertices, which are introduced in Imran et al. (2014, Ars. Combinatoria 117, 113-130), are instrumental in determining the lower bound of the metric dimension for certain types of graphs. We propose an approach, based on these notions, to determine the lower bound of $\dim(P(n,3))$. Moreover, we shall prove that $\dim(P(n,3))=4$, where $n\equiv2,3,4,5 \,\,(\text{mod}\,\, 6)$ and is sufficiently large.