On the Bounded Partition Dimension of Some Generalised Graph Structures

Abstract

Consider λ to be a connected graph with a vertex set Vλ that may be partitioned into any partition set S. If each vertex in λ has a separate representation with regard to S and is an ordered k partition, then the set with S is a resolving partition of λ.. A partition dimension of λ, represented by pd, is the minimal cardinality of resolving k partitions of Vλ. The partition dimension of various generalised families of graphs, such as the Harary graph, Cayley graph, and Pendent graph, is given as a sharp upper bound in this article.