  A directed dominating set in a directed graph $D$ is a set $S$ of vertices of
$V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$
in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted
by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$.
The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is
the maximum directed domination number $\gamma(D)$ over all orientations $D$ of
$G$. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if $\alpha$ denotes the independence number of a graph
$G$, we show that $\alpha \le \Gamma_d(G) \le \alpha(1+2\ln(n/\alpha))$.

Combinatorics