On Sharp Bounds on Partition Dimension of Convex Polytopes

Abstract

Let $\Omega $ be a connected graph and for a given $l$ -ordered partition of vertices of a connected graph $\Omega $ is represented as $\beta =\{\beta _{1},\beta _{2}, {\dots },\beta _{l}\}$ . The representation of a vertex $\mu \in V(\Omega)$ is the vector $r(\mu |\beta)=(d(\mu,\beta _{1}),d(\mu,\beta _{2}), {\dots }, d(\mu,\beta _{l}))$ . The partition $\beta $ is a resolving partition for vertices of $\Omega $ if all vertices of $\Omega $ having the unique representation with respect to $\beta $ . The minimum number of $l$ in the resolving partition for $\Omega $ is known as the partition dimension of $\Omega $ and represented as $pd(\Omega)$ . Resolving partition and partition dimension have multipurpose applications in networking, optimization, computer, mastermind games and modeling of chemical structures. The problem of computing constant values of partition dimension is NP-hard so one can find sharp bound for the partition dimension of graph. In this article, we computed the upper bound for the convex polytopes $\mathbb {E}_{n},~\mathbb {S}_{n},~\mathbb {T}_{n},~\mathbb {G}_{n},~\mathbb {Q}_{n}$ and flower graph $f_{n\times 3}$ .

Keywords

• Resolving partition
• partition dimension
• convex polytope
• flower graph