On Sharp Bounds on Partition Dimension of Convex Polytopes

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
$f_{n\times 3}$
.