On Metric Dimension of Convex Polytopes with Pendant Edges

Abstract

A family \(\mathcal{G}\) of connected graphs is said to be a family with constant metric dimension if \(\dim(G)\) does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of some plane graphs obtained from convex polytopes by attaching a pendant edge to each vertex of the outer cycle in a plane representation of these convex polytopes. We prove that the metric dimension of these plane graphs is constant and only three vertices, appropriately chosen, suffice to resolve all vertices of these classes of graphs. It is natural to ask for the characterization of graphs \(G\) that are plane representations of convex polytopes having the property that \(\dim(G) = \dim(G’)\), where \(G’\) is obtained from \(G\) by attaching a pendant edge to each vertex of the outer cycle of \(G\).