  In a digraph, a quasi-kernel is a subset of vertices that is independent and
such that every vertex can reach some vertex in that set via a directed path of
length at most two. Whereas Chv\'atal and Lov\'asz proved in 1974 that every
digraph has a quasi-kernel, very little is known so far about the complexity of
finding small quasi-kernels. In 1976 Erd\H{o}s and Sz\'ekely conjectured that
every sink-free digraph $D = (V, A)$ has a quasi-kernel of size at most
$|V|/2$. Obviously, if $D$ has two disjoint quasi-kernels then it has a
quasi-kernel of size at most $|V|/2$, and in 2001 Gutin, Koh, Tay and Yeo
conjectured that every sink-free digraph has two disjoint quasi-kernels. Yet,
they constructed in 2004 a counterexample, thereby disproving this stronger
conjecture. We shall show that, not only sink-free digraphs occasionally fail
to contain two disjoint quasi-kernels, but it is computationally hard to
distinguish those that do from those that do not. We also prove that the
problem of computing a small quasi-kernel is polynomial time solvable for
orientations of trees but is computationally hard in most other cases (and in
particular for restricted acyclic digraphs).
