Directed domination in oriented hypergraphs

Erd\H{o}s [On Sch"utte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $\log (n+1)$vertices,where$\log$isthelogarithmtobase$2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $\log n-2\log \log n+1$. This notion of directed domination number has been generalized to arbitrary graphs by Caro and Henning in [Directed domination in oriented graphs, Discrete Appl. Math. (2012) 160:7–8.]. However, the generalization to directed r-uniform hypergraphs seems to be rare. Among several results, we prove the following upper and lower bounds on ${\overrightarrow{\mathrm{\Gamma }}}_{r-1}(H(n,r))$, the upper directed $(r-1)$-domination number of the complete $r$-uniform hypergraph on $n$ vertices $H(n,r)$, which is the main theorem of this paper: $$c(\ln n{)}^{\frac{1}{r-1}}\le {\overrightarrow{\mathrm{\Gamma }}}_{r-1}(H(n,r))\le C\ln n,$$ where $r$ is a positive integer and $c=c(r)>0$ and $C=C(r)>0$ are constants depending on $r$.