On the partition dimension of trees

Given an ordered partition Π={P1,P2,…,Pt} of the vertex set V of a connected graph G=(V,E), the partition representation of a vertex v∈V with respect to the partition Π is the vector r(v|Π)=(d(v,P1),d(v,P2),…,d(v,Pt)), where d(v,Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have different partition representations, i.e., for every pair of vertices u,v∈V, r(u|Π)≠r(v|Π). The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we obtain several tight bounds on the partition dimension of trees.