On the k-domination number of digraphs

Let $$k\ge 1$$k≥1 be an integer and let D be a digraph with vertex set V(D). A subset $$S\subseteq V(D)$$S⊆V(D) is called a k-dominating set if every vertex not in S has at least k predecessors in S. The k-domination number $$\gamma _{k}(D)$$γk(D) of D is the minimum cardinality of a k-dominating set in D. We know that for any digraph D of order n, $$\gamma _{k}(D)\le n$$γk(D)≤n. Obviously the upper bound n is sharp for a digraph with maximum in-degree at most $$k-1$$k-1. In this paper we present some lower and upper bounds on $$\gamma _{k}(D)$$γk(D). Also, we characterize digraphs achieving these bounds. The special case $$k=1$$k=1 mostly leads to well known classical results.