## On The Partition Dimension of Disconnected Graphs

Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung 40132, Indonesia.

Received: May 20th, 2016, 1st Revision: September 5th, 2016, Accepted to Publish: September 16th, 2016.

Abstract. For a graph G = (V;E); a partition Ω = {O1,O2,...,Ok} of V(G) of the vertex set V is called a resolving partition if every pair of vertices u,v ϵ V(G) have distinct representations under Ω: The partition dimension of G is the minimum integer k such that G has a resolving k-partition. Many results in determining the partition dimension of graphs have been obtained. However, known results are only limited for connected graphs. In this paper, we extend the notion of the partition dimension of a graph so that we could apply it to disconnected graphs as well. We determine some lower and upper bounds for the partition dimension of a disconnected graph (if they are finite). We also give the partition dimensions for some classes of disconnected graphs.

Keywords : resolving partition; partition dimension; disconnected graph.

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