  An independent vertex subset $S$ of the directed graph $G$ is a kernel if the
set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset
$Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors
contains $V(G)\setminus S$ as a subset. Deciding whether a directed graph has a
kernel is an NP-hard problem. In stark contrast, each directed graph has
quasi-kernel(s) and one can be found in linear time. In this article, we will
survey the results on quasi-kernel and their connection with kernels. We will
focus on the small quasi-kernel conjecture which states that if the graph has
no vertex of zero in-degree, then there exists a quasi-kernel of size not
larger than half of the order of the graph. The paper also contains new proofs
and some new results as well.
