On the Metric Dimension of Generalized Petersen Graphs

Abstract

A family of connected graphs \(\mathcal{G}\) is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of the generalized Petersen graphs \(P(n,m)\) for \(n = 2m+1\) and \(m \geq 1\) and give a partial answer to the question raised in \([9]\): Is \(P(n, m)\) for \(n \geq 7\) and \(3 \leq m \leq \lfloor \frac{n-1}{2} \rfloor\) a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs \(P(n,m)\) with \(n = 2m +1\) have metric dimension \(3\) for every \(m \geq 2\).