Non-Isolated Resolving Number of Graph with Pendant Edges

Abstract

We consider V; E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = fw g of vertices and a vertex v 2 G, the ordered r(vjW) = (d(v; w 1 ); d(v; w 2 1 ; w 2 ; w ); : : : ; d(v; w 3 ; : : : ; w k k )) of k-vector is representations of v with respect to W, where d(v; w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of jWj. Furthermore, the resolving set W of graph G is called nonisolated resolving set if there is no 8v 2 W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges.