On the k-partition dimension of graphs

As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G=(V,E), a partition II of V is said to be a k-partition generator of G if any pair of different vertices u,v E V is distinguished by at least k vertex sets of II i.e., there exist at least k vertex sets S1,...,Sk E II such that d(u,Si) /= d(v,Si) for every i E {1,...,k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k E {1,...,r}