On Partition Dimension of Generalized Convex Polytopes

Abstract

Let $\mathbb{G}$ be a graph having no loop or multiple edges, $k-$ order vertex partition for $\mathbb{G}$ is represented by $\gamma =\left\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_k\right\}$ . The vector $r\left(\varphi \left|\gamma \right.\right)=\left(d\left(\varphi ,{\gamma }_1\right),d\left(\varphi ,{\gamma }_2\right),d\left(\varphi ,{\gamma }_3\right)\cdots ,d\left(\varphi ,{\gamma }_k\right)\right)$ is the representation of vertex $\varphi$ with respect to $\gamma$ . If the representation of all the vertices with respect to $\gamma$ is different, then $\gamma$ is said to be resolving partition for the graph $\mathbb{G}$ . The minimum number $k$ is resolving partition for $\mathbb{G}$ and is termed as partition dimension for $\mathbb{G}$ , represented by $pd\left(\mathbb{G}\right)$ . There are numerous applications of partition dimension in different fields such as optimization, computer, mastermind games, and networking and also in modeling of numerical structure. The problem of finding constant value of partition dimension for a graph or network is very hard, so one can find bounds for the partition dimension. In this work, we consider convex polytopes in their generalized forms that are $E_n,S_n$ , and $G_n$ , and we compute upper bounds for the partition dimension of the desired polytopes.