Greedily Constructing Small Quasi-Kernels

In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chv\'atal and Lov\'asz showed every digraph contains a quasi-kernel. In 1976, P. L. Erd\H{o}s and Sz\'ekely conjectured that every sourceless digraph has a quasi-kernel of order at most $\frac{n}{2}$. Despite significant recent attention by the community the problem remains far from solved, with no bound of the form $(1-\epsilon)n$ known. We introduce a polynomial time algorithm which greedily constructs a small quasi-kernel. Using this algorithm we show that if $D$ is a $\vec{K}_{1,d}$-free digraph, then $D$ has a quasi-kernel of order at most $\frac{(d^2 - 2d + 2)n}{d^2-d+1}$. By refining this argument we prove that for any $D$ with maximum out-degree $3$ this algorithm constructs a quasi-kernel of order at most ${4n}/{7}$. Finally, we consider the problem in digraphs forbidding certain orientation of short cycles as subgraphs, concluding that all orientations $D$ of a graph $G$ with girth at least $7$ have a quasi-kernel of order at most $\frac{(d^2+4)n}{(d+2)^2}$, where $d$ is the maximum out-degree of $D$.