### 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.

**How to Cite this Article:**B. Sooryanarayana, K.N. Geetha, On the k-metric dimension of graphs, J. Math. Comput. Sci., 4 (2014), 861-878 Copyright © 2014 B. Sooryanarayana, K.N. Geetha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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