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 $$\gamma _{k}(D)$$ γ k ( D ) of D is the minimum cardinality of a k-dominating set in D. We know that for any digraph D of order n, $$\gamma _{k}(D)\le n$$ γ k ( 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 $$\gamma _{k}(D)$$ γ k ( D ) . Also, we characterize digraphs achieving these bounds. The special case $$k=1$$ k = 1 mostly leads to well known classical results.