On Some Plane Graphs and Their Metric Dimension

Abstract

Metric basis and metric dimension has become an integral part of molecular topology and combinatorial chemistry. It has a lot of applications in pharmacy, chemistry, computer, and mathematical disciplines. Let $\Phi =(V,E)$ be a simple connected graph. A subset $F=\{y_{1}, y_{2}, y_{3},\ldots ,y_{s}\}$ of distinct vertices from $V(\Phi )$ is called a resolving set if for any $a \in V(\Phi )$, the metric code of a with respect to F that is denoted by $\zeta (a|F)$, which is defined as $\zeta (a|F)=(d(y_{1},a), d(y_{2},a), d(y_{3}, a),\ldots , d(y_{s},a))$, is distinct for distinct a. Then, such a set F with minimum cardinality is called the metric basis for $\Phi$, and this minimum cardinality of a resolving set F is called the metric dimension of $\Phi$ and is denoted by $dim(\Phi )$. A polytope in elementary geometry is a geometric object with flat sides. The polytopes which are convex sets and are contained in the n-dimensional space $\mathbb {R}^{n}$ (Euclidean space) are termed convex polytopes. Convex polytopes have found a lot of applications in different areas of computer science and mathematics. In this work, we construct two closely related families of convex polytope graphs (viz., $\mathfrak {D}_{n}$ and $\mathfrak {Q}_{n}$), using the existing families of convex polytope graphs. We study the metric dimension for these graphs and prove that only three vertices are a minimum requirement for the unique identification of all vertices in these graphs. We also give some partial answers to the problems raised in the recent past.