  A non-empty subset $S$ of the vertices of a digraph $D$ is called a {\it safe
set} if \begin{itemize}
  \item[(i)] for every strongly connected component $M$ of $D-S$, there exists
a strongly connected component $N$ of $D[S]$ such that there exists an arc from
$M$ to $N$; and \item[(ii)] for every strongly connected component $M$ of $D-S$
and every strongly connected component $N$ of $D[S]$, we have $|M|\leq |N|$
whenever there exists an arc from $M$ to $N$. \end{itemize} In the case of
acyclic digraphs a set $X$ of vertices is a safe set precisely when $X$ is an
{\it in-dominating set}, that is, every vertex not in $X$ has at least one arc
to $X$. We prove that, even for acyclic digraphs which are traceable (have a
hamiltonian path) it is NP-hard to find a minimum cardinality in-dominating
set. Then we show that the problem is also NP-hard for tournaments and give,
for every positive constant $c$, a polynomial algorithm for finding a minimum
cardinality safe set in a tournament on $n$ vertices in which no strong
component has size more than $c\log{}(n)$. Under the so called Exponential Time
Hypothesis (ETH) this is close to best possible in the following sense: If ETH
holds, then, for every $\epsilon>0$ there is no polynomial time algorithm for
finding a minimum cardinality safe set for the class of tournaments in which
the largest strong component has size at most $\log^{1+\epsilon}(n)$.
  We also discuss bounds on the cardinality of safe sets in tournaments.
