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

S. M. Yunika (CGANT University of Jember Mathematics Depart. University of Jember)
Slamin Slamin (CGANT University of Jember System Information. Depart. University of Jemb)
Dafik Dafik (CGANT University of Jember Mathematics Edu. Depart. University of Jember)
Kusbudiono Kusbudiono (CGANT University of Jember Mathematics Depart. University of Jember)

UNEJ e-Proceeding
2016: Proceeding The 1st International Basic Science Conference

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.