On the partition dimension of unicyclic graphs

الأصل · EN

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 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 for G. In this paper we obtain several tight bounds on the partition dimension of unicyclic graphs.