A vertex set
![$X$](Abs/img1.gif)
of a digraph
![$D=(V,A)$](Abs/img2.gif)
is a
kernel if
![$X$](Abs/img1.gif)
is independent (i.e., all pairs of distinct vertices of
![$X$](Abs/img1.gif)
are
non-adjacent) and for every
![$v\in V-X$](Abs/img3.gif)
there exists
![$x\in X$](Abs/img4.gif)
such that
![$vx\in
A$](Abs/img5.gif)
. A vertex set
![$X$](Abs/img1.gif)
of a digraph
![$D=(V,A)$](Abs/img2.gif)
is a
quasi-kernel if
![$X$](Abs/img1.gif)
is
independent and for every
![$v\in V-X$](Abs/img3.gif)
there exist
![$w\in V-X, x\in X$](Abs/img6.gif)
such that
either
![$vx\in
A$](Abs/img5.gif)
or
![$vw,wx\in A.$](Abs/img7.gif)
In 1994, Chvátal and Lovász proved that
every digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that, if
a digraph
![$D$](Abs/img8.gif)
has no kernel, then
![$D$](Abs/img8.gif)
contains at least three quasi-kernels.
We characterize digraphs with exactly one and two quasi-kernels, and, thus,
provide necessary and sufficient conditions for a digraph to have at least
three quasi-kernels. In particular, we prove that every strong digraph of
order at least three, which is not a 4-cycle, has at least three
quasi-kernels. We conjecture that every digraph with no sink has a pair of
disjoint quasi-kernels and provide some support to this
conjecture