On the metric dimension of bipartite graphs

For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in connected graph G, the k-vector r(v|W)=(d(v,w1),d(v,w2),…,d(v,wk)) is called representation with respect to W, where d(v,wi) distance between wi, for 1≤i≤k. The set W resolving G if r(u|W)≠r(v|W), every pair u,v∈V(G). minimum positive integer k which has cardinality metric dimension denoted as dim(G). A dim(G) basis G. bipartite projection defined one partite sets, two are adjacent they have at least common neighbor other set. In this paper, we investigate relation its projections. Furthermore, present some realization results bounds on graph.