Star metric dimension of complete, bipartite, complete bipartite and fan graphs

Abstract

One of the topics in graph theory that is interesting and developed continuously is metric dimension.  It has some new variation concepts, such as star metric dimension. The order set of Z={z1,z2,…,zn }⊆V(G) called star resolving set of connected graph  G if Z is Star Graph and for every vertex in G has different representation to the set Z. The representation is expressed as the distance d(u,z), it is the shortest path from vertex u to z for every u,z∈V(G). Star basis of a graph is the smallest cardinality of star resolving set. The number of vertex in star basis is called star metric dimension of G which denoted by Sdim(G). The purpose of this article is to determine the characteristic of star metric dimension and the value of star metric dimension of some classes of graphs. The method which is used in this study is library research. Some of the results of this research are complete graph has  Sdim(Kn )=n-1, for n≥3, bipartite graph K(2,n)  has  Sdim (K(2,n) )=n, for n≥3. Besides complete bipartite graph hasn’t star metric dimension or  for m,n≥3 or it can said that  Sdim (K(m,n) )=0. Another graph, that is Fan graph has Sdim (Fn )=2 for 2≤n≤5 and for n≥6 Sdim (Fn )=(2n+3)/5.