Some new results on the k-tuple domination number of graphs

Let k ≥ 1 be an integer and G be a graph of minimum degree δ(G) ≥ k − 1. A set D ⊆ V(G) is said to be a k-tuple dominating set of G if |N[v] ∩ D| ≥ k for every vertex v ∈ V(G), where N[v] represents the closed neighbourhood of vertex v. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this paper, we continue with the study of this classical domination parameter in graphs. In particular, we provide some relationships that exist between the k-tuple domination number and other classical parameters, like the multiple domination parameters, the independence number, the diameter, the order and the maximum degree. Also, we show some classes of graphs for which these relationships are achieved.