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.