On the Bounded Partition Dimension of Some Generalised Graph Structures

Abstract

Consider $\lambda$ to be a connected graph with a vertex set $V\left(\lambda \right)$ that may be partitioned into any partition set $S.$ If each vertex in $\lambda$ has a separate representation with regard to $S$ and is an ordered $k$ partition, then the set with $S$ is a resolving partition of $\lambda .$ . A partition dimension of $\lambda ,$ represented by $pd,$ is the minimal cardinality of resolving $k$ partitions of $V\left(\lambda \right).$ 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.