Resolving Sets without Isolated Vertices

Let G be a connected graph. Let W = (w1, w2, ..., wk ) be a subset of V with an order imposed on it. For any v ∈ V, the vector r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wk )) is called the metric representation of v with respect to W. If distinct vertices in V have distinct metric representations, then W is called a resolving set of G. The minimum cardinality of a resolving set of G is called the metric dimension of G and it is denoted by dim(G). A resolving set W is called a non-isolated resolving set if the induced subgraph (W) has no isolated vertices. The minimum cardinality of a non-isolated resolving set of G is called the non-isolated resolving number of G and is denoted by nr(G). In this paper, we initiate a study of this parameter.