A generalization of independent resolving partition of a graph

Let G = (V, E) be a simple connected graph. A partition ᴨ = {V1, V2, V3….,Vk } is called a resolving partition of G if for any u ϵ V(G) , the code of u with respect to ᴨ (denoted by cᴨ (u) ) namely (d(u,V1 ),d(u, V2),….,d(u, Vk )) is distinct for different u ϵ V(G) where
d(u, Vi)= min{d(u, x) / x ϵ Vi }. The minimum cardinality of a resolving partition of a graph G is called the partition dimension of G and is denoted by pd (G)[2]. Several types of resolving partition have been considered like connected resolving partition [7], metric chromatic number of a graph (that is, independent resolving partition) [4], equivalence resolving partition [6] etc. A new type of resolving partition called isolate vertex resolving partition was introduced in [5].This partition is a generalization of an independent resolving partition. A detailed study of this partition is done in this paper.