On the k-metric dimension of graphs

Abstract

Let G(V,E) be a connected graph. A subset S ofV is said to be k-resolving set of G, if for every pair of distinct vertices u,v / ∈ S, there exists a vertex w ∈ S such that |d(u,w)−d(v,w)| ≥ k, for some k ∈ Z+. Among all k-resolving sets of G, the set having minimum cardinality is called a k-metric basis of G and its cardinality is called the k-metric dimension of G and is denoted by βk(G). In this paper, we have discussed some characterizations of k-metric dimension in terms of some graphical parameters. We have mainly focused on 2-metric dimension of graphs and discussed few characterizations. Further 2-metric dimension of trees is determined and from this result 2- metric dimension of path, cycle and sharp bounds of unicyclic graphs are established.