# Directed domination in oriented hypergraphs

## TL;DR

This paper extends the concept of directed domination from graphs to hypergraphs, establishing bounds on the directed domination number of complete r-uniform hypergraphs, thus broadening understanding in hypergraph domination theory.

## Contribution

It provides the first known bounds on the upper directed (r-1)-domination number for complete r-uniform hypergraphs, generalizing classical results from tournaments to hypergraphs.

## Key findings

- Upper bound of C ln n for the directed domination number
- Lower bound of c (ln n)^{1/(r-1)} for the directed domination number
- Main theorem establishing these bounds depending on r

## Abstract

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 $\log$ is the logarithm to base $2$. 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 $\overrightarrow{\Gamma}_{r-1}(H(n,r))$, 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: \[c (\ln n)^{\frac{1}{r-1}} \le \overrightarrow{\Gamma}_{r-1}(H(n,r)) \le C \ln n,\] where $r$ is a positive integer and $c= c(r) > 0$ and $C = C(r) > 0$ are constants depending on $r$.