The Small Quasi-Kernel Conjecture

Let $D$ be a sourceless digraph on $n$ vertices. A quasi-kernel is an independent set $Q$ such that every vertex $u$ of $D$ has a vertex $v\in Q$ with $\operatorname{dist}(v,u)\leq 2$.

Small Quasi-Kernel Conjecture. The digraph $D$ contains a quasi-kernel of order at most $\frac{n}{2}$.

Sources without vertices have no immediate obstruction from isolated source vertices to a small quasi-kernel, motivating this conjecture of Erdős and Székely from 1976. The conjecture was initially communicated orally and later appeared in print; its status is not resolved in the supplied text.