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 = {Rp1, Rp2, . . . , Rpk} and the representation r(a|Rp) of a vertex a with respect to Rp is the vector (d(a|Rp1), d(a|Rp2), . . . , d(a|Rpk)). The partition is called a resolving partition of G if r(a|Rp) # r(b|Rp) for all distinct a, b is an element of 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 Tn, Un, Vn, and An, and proved that these graphs have constant partition dimension.