On the metric dimension of rotationally-symmetric convex polytopes

Year 2016, Volume: 3 Issue: 2, 45 - 59, 15.05.2016

Muhammad Imran , Syed Ahtsham Ul Haq Bokhary , A. Q. Baig

https://doi.org/10.13069/jacodesmath.47485

Cited By: 4

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.

Keywords

Metric dimension, Basis, Resolving set, Prism, Antiprism, Convex polytopes

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