On the k-domination number of digraphs

Abstract

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