On Vertex-Based Dimension of Some Graphs Joining Certain Prism Graphs

Abstract

Background: In graph theory, the prism graph is a type of graph that is characterised by having the structure of a prism as its underlying framework. The notion of a resolving set and that of metric dimension for a graph of a prism is important in uniquely identifying the vertices within a prism graph. For a non-trivial connected graph $\Gamma_{r}=\Gamma_{r}(V, E)$, an ordered subset $U$ of vertices $resolves$ any pair of different vertices $y_{1}, y_{2} \in V$, if $d(v, y_{1})\neq d(v, y_{2})$ for some $v\in U$. Such a set $U$ is said to be a resolving set for $\Gamma_{r}$ and the smallest cardinality of $U$ is called the $metric$ $dimension$ of $\Gamma_{r}$. 
Purpose: The purpose of this article is to determine the notion of resolving sets and their corresponding metric dimensions for two complex families of planar graphs obtained by joining $m-$copies of the prism graph on known families of convex polytope graphs.
Methods: The methods used are purely theoretical, based on mathematical reasoning and established definitions related to graph theory.
Results: In this article, we have determined successfully the resolving set and metric dimension for two specific complex families of planar graphs, denoted by $\mathrm{L}_{n}$ and $\mathrm{M}_{n}$, constructed using $m-$copies of a prism graph. These findings contribute to our understanding of these concepts within graph theory.
Conclusions: This research indicates the importance of studying resolving sets and metric dimensions in various graph structures, particularly those derived from multiple copies of the prism graph connected through known families of convex polytope graphs. This work may inspire further investigations into similar graph families or other applications of these concepts in different areas of mathematics and computer science.

Keywords: Metric dimension, independent set, basis set, convex polytope, prism graphs
2000 Mathematics Subject Classification: 05C12