unicyclic graphs with non-isolated resolving number 2

Let g be a connected graph and w = {w1, w2, . . . , wk} be an ordered subset of vertices of g. for any vertex v of g, the ordered k-vector r(v|w ) = (d(v, w1), d(v, w2), . . . , d(v, wk )) is called the metric representation of v with respect to w , where d(x, y) is the distance between the vertices x and y. a set w is called a resolving set for g if distinct vertices of g have distinct metric representations with respect to w . the minimum cardinality of a resolving set for g is its metric dimension denoted by dim(g). a resolving set w is called a non-isolated resolving set for g if the induced subgraph (w) of g has no isolated vertices. the minimum cardinality of a non-isolated resolving set for g is called the non-isolated resolving number of g and denoted by nr(g). the aim of this paper is to find properties of unicyclic graphs that have non-isolated resolving number 2 and then to characterize all these graphs.