A Special Class of Convex Polytopes with Constant Metric Dimension

Abstract

A family $\mathcal{G}$ of connected graphs is a family with constant metric dimension if $\dim (\mathcal{G})$ is finite and does not depend upon the choice of $G$ in $\mathcal{G}$.

The metric dimension of some classes of convex polytopes has been determined in $[8-12]$ and an open problem was raised in $[10]$: \emph{Let $G$ be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes $G_1$ and $G_2$ (such that the outer cycle of $G_1$ is the inner cycle of $G_2$) both having constant metric dimension. Is it the case that $G$ will always have the constant metric dimension?}

In this paper, we study the metric dimension of an infinite class of convex polytopes which are obtained by the combinations of two different graphs of convex polytopes. It is shown that this infinite class of convex polytopes has constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes.