On Sharp Bounds on Partition Dimension of Convex Polytopes
Yu‐Ming Chu, Muhammad Faisal Nadeem, Muhammad Sohaib Azeem, Muhammad Kamran Siddiqui · IEEE Access · 2020
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}$.