On the partition dimension of disconnected graphs

Abstract

For a graph G = (V,E), a partition ω = {01,02,.., Ok] of the vertex set V is called a resolving partition if every pair of vertices u, v ∈ V(G) have distinct representations under ω. The partition dimension of G is the minimum integer k such that G has a resolving fc-partition. Many results in determining the partition dimension of graphs have been obtained. However, the known results are limited to connected graphs. In this study, the notion of the partition dimension of a graph is extended so that it can be applied to disconnected graphs as well. Some lower and upper bounds for the partition dimension of a disconnected graph are determined (if they are finite). In this paper, also the partition dimensions for some classes of disconnected graphs are given.