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 Pi of V is said to be a k-partition generator of G if any pair of different vertices u, v is an element of V is distinguished by at least k vertex sets of Pi, i.e., there exist at least k vertex sets S1, ..., S-k is an element of Pi such that d(u, S-i) not equal d(v, S-i) for every i is an element of(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 is an element of{1, ..., r}. (C) 2018 Elsevier B.V. All rights reserved.