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

Abstract

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 $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ 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|{R}{p}) \ne r(b|{R}{p}) $ 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.

Keywords

• convex polytope-like graph
• bounded partition dimension
• partition dimension
• partition resolving set