On the metric dimension of rotationally-symmetric convex polytopes

Abstract

Metric dimension is a~generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let $\mathcal{F}$ be a family of connected graphs $G_{n}$ : $\mathcal{F} = (G_{n})_{n}\geq 1$ depending on $n$ as follows: the order $|V(G)| = \varphi(n)$ and $\lim\limits_{n\rightarrow \infty}\varphi(n)=\infty$. If there exists a constant $C > 0$ such that $dim(G_{n}) \leq C$ for every $n \geq 1$ then we shall say that $\mathcal{F}$ has bounded metric dimension, otherwise $\mathcal{F}$ has unbounded metric dimension. If all graphs in $\mathcal{F}$ have the same metric dimension, then $\mathcal{F}$ is called a family of graphs with constant metric dimension.\\ In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.