Unicyclic graphs with non-isolated resolving number ۲

Let G be a connected graph and W=\{w_۱, w_۲,\ldots,w_k\} be an ordered subset of vertices of G. For any vertex v of G, the ordered k-vector r(v|W)=(d(v,w_۱), d(v,w_۲),\ldots,d(v,w_k)) 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 \langle W\rangle 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 ۲ and then to characterize all these graphs.

کلیدواژه ها:

non-isolated resolving sets ، Unicyclic graphs ، Metric dimension