Cardinality bounds on subsets in the partition resolving set for complex convex polytope-like graph

Let $G=(V,E)$ be a simple, connected graph with vertex set $V(G)$ and $E(G)$ edge set of $G$. For two vertices $a$ and $b$ in a graph $G$, the distance $d(a,b)$ from $a$ to $b$ is the length of shortest path $a-b$ path in $G$. A $k$-ordered partition of vertices of $G$ is represented as $Rp=\{R{p}_1,R{p}_2,\dots ,R{p}_k\}$ and the representation $r(a|Rp)$ of a vertex $a$ with respect to $Rp$ is the vector $(d(a|R{p}_1),d(a|R{p}_2),\dots ,d(a|R{p}_k))$. The partition is called a resolving partition of $G$ if $r(a|Rp)\ne r(b|Rp)$ for all distinct $a,b\in V(G)$. The partition dimension of a graph, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically ${\mathbb{T}}_n$, ${\mathbb{U}}_n$, ${\mathbb{V}}_n$, and ${\mathbb{A}}_n$, and proved that these graphs have constant partition dimension.