On metric dimension of plane graphs $\mathfrak{J}_{n}$, $\mathfrak{K}_{n}$ and $\mathfrak{L}_{n}$

Abstract

Let $\Gamma=\Gamma(\mathbb{V},\mathbb{E})$ be a simple (i.e., multiple edges and loops and are not allowed), connected (i.e., there exists a path between every pair of vertices), and an undirected (i.e., all the edges are bidirectional) graph. Let $d_{\Gamma}(\varrho_{i},\varrho_{j})$ denotes the geodesic distance between two nodes $\varrho_{i},\varrho_{j} \in \mathbb{V}$. The problem of characterizing the classes of plane graphs with constant metric dimensions is of great interest nowadays. In this article, we characterize three classes of plane graphs (viz., $\mathfrak{J}_{n}$, $\mathfrak{K}_{n}$, and $\mathfrak{L}_{n}$) which are generated by taking n-copies of the complete bipartite graph (or a star) $K_{1,5}$, and all of these plane graphs are radially symmetrical with the constant metric dimension. We show that three vertices is a minimal requirement for the unique identification of all vertices of these three classes of plane graphs.