Bounds on the domination number of a digraph

Abstract

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S is adjacent from at least one vertex in S. The domination number of D, denoted by $\gamma (D)$, is the minimum cardinality of a dominating set of D. The Slater number $s\ell (D)$ is the smallest integer t such that t added to the sum of the first t terms of the non-increasing out-degree sequence of D is at least as large as the order of D. For any digraph D of order n with maximum out-degree $\Delta ^+$, it is known that $\gamma (D)\ge \lceil n/(\Delta ^++1)\rceil$. We show that $\gamma (D)\ge s\ell (D)\ge \lceil n/(\Delta ^++1)\rceil$ and the difference between $s\ell (D)$ and $\lceil n/(\Delta ^++1)\rceil$ can be arbitrarily large. In particular, for an oriented tree T of order n with $n_0$ vertices of out-degree 0, we show that $(n-n_0+1)/2\le s\ell (T)\le \gamma (T)\le 2s\ell (T)-1$ and moreover, each value between the lower bound $s\ell (T)$ and the upper bound $2s\ell (T)-1$ is attainable by $\gamma (T)$ for some oriented trees. Further, we characterize the oriented trees T for which $s\ell (T)=(n-n_0+1)/2$ hold and show that the difference between $s\ell (T)$ and $(n-n_0+1)/2$ can be arbitrarily large. Some other elementary properties involving the Slater number are also presented.