On The Partition Dimension of Disconnected Graphs

Abstract

Abstract. For a graph, , a partition Ω , , … , of the vertex set is called a resolving partition if every pair of vertices , ∈ have distinct representations under Ω. The partition dimension of is the minimum integer such that has a resolving -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.

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

“…In [38] bounds of uni-cyclic graphs are presented. For more recent literature and results, we refer to [13], [17], [20], [26], [28], [30]- [32], [34], [37], [38].…”

A method to construct graphs with certain partition dimension

“…Further results for the metric dimension of graphs can be seen in [1,2,3,4,14,15]. In 2015, the notion of the partition dimension of a graph was generalized by Haryeni et al [12,13] so that the notion can be applied to all graphs (connected as well as disconnected graphs).…”

“…For a disconnected graph G = m i=1 G i , Haryeni et al [12] derived the upper and lower bounds of the partition dimension of G (if it is finite), namely…”