On The Metric Dimension with Non-isolated Resolving Number of Some Exponential Graph

Abstract

Let w, w ∈ G = (V, E). A distance in a simple, undirected and connected graph G, denoted by d(v, w), is the length of the shortest path between v and w in G. For an ordered set W = {w1, w2, w3, . . . , wk} of vertices and a vertex v ∈ G, the ordered k-vector r(v|W) = (d(v, w1), d(v, w2), . . . , d(v, wk)) is representations of v with respect to 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 dim(G) of G is the minimum cardinality of resolving set for G. The resolving set W of graph G is called non-isolated resolving set if subgraph W is induced by non-isolated vertex. While the minimum cardinality of non-isolated resolving set in graph is called a non-isolated resolving number, denoted by nr(G). In this paper we study a metric dimension with non-isolated resolving number of some exponential graph.