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arXiv · 2508.12224

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

Rui Gao·Yingqing Xiao·Zhanqi Zhang
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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.

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Rui Gao, Yingqing Xiao, Zhanqi Zhang. 2025-08-17. On the Metric Dimension of Generalized Petersen Graphs $P(n,3)$. https://arxiv.org/abs/2508.12224

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