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

Abstract

In this paper, we investigate the metric dimension of generalized Petersen graphs \(P(n,3)\), providing a partial answer to an open problem posed in [8]: whether \(P(n,m)\) for \(n \geq 7\) and \(3 \leq m \leq \left\lfloor \frac{n-1}{2} \right\rfloor\) constitutes a family of graphs with constant metric dimension. Specifically, we prove that the metric dimension of \(P(n,3)\) equals \(3\) for \(n \equiv 1 \pmod{6}\), \(n \geq 25\), and equals \(4\) for \(n \equiv 0 \pmod{6}\), \(n \geq 24\). For remaining cases, four judiciously chosen vertices suffice to resolve all vertices of \(P(n,3)\), implying \(\dim(P(n,3)) \leq 4\), except when \(n \equiv 2 \pmod{6}\), in which case \(\dim(P(n,3)) \leq 5\).