A vertex set

of a digraph

is a
kernel if

is independent (i.e., all pairs of distinct vertices of

are
non-adjacent) and for every

there exists

such that

. A vertex set

of a digraph

is a
quasi-kernel if

is
independent and for every

there exist

such that
either

or

In 1994, Chvátal and Lovász proved that
every digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that, if
a digraph

has no kernel, then

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