On the metric dimension of bipartite graphs

Abstract

For an ordered subset W = { w 1 , w 2 , … , w k } of vertices and a vertex v in a connected graph G, the ordered k-vector r ( v | W ) = ( d ( v , w 1 ) , d ( v , w 2 ) , … , d ( v , w k ) ) is called the representation of v with respect to W, where d ( v , w i ) is the distance between v and wi , for 1 ≤ i ≤ k . The set W is called a resolving set of G if r ( u | W ) ≠ r ( v | W ) , for every pair u , v ∈ V ( G ) . The minimum positive integer k for which G has a resolving set of cardinality k is the metric dimension of G, denoted as dim(G). A resolving set of G of cardinality dim(G) is a metric basis of G. For a bipartite graph G, projection is defined as a graph with the vertices of one of the partite sets, where two vertices are adjacent if they have at least one common neighbor in the other partite set. In this paper, we investigate the relation between the metric dimension of a bipartite graph and its projections. Furthermore, we present some realization results for the bounds on the metric dimension of a bipartite graph.