Small $q$-kernels in digraphs with minimum in-degree $\delta$

For a digraph $D$, a subset $Q\subseteq V(D)$ is called a $q$-kernel if $Q$ is an independent set and all vertices in $V(D)$ are reachable from $Q$ via a directed path of length at most $q$. Given integers $q\ge 2$ and $\delta \ge 1$, Spiro arXiv:2404.07305 [math.CO] posed the question: what is the smallest constant $c_{\delta ,q}$ such that every digraph $D$ with minimum in-degree $\delta$ has a $q$-kernel of size at most $c_{\delta ,q}\mathrm{\mid }V(D)\mathrm{\mid }$? We show the constants $c_{\delta ,q}$ are monotone in both $\delta$ and $q$, and we improve upon the known upper bounds for $c_{\delta ,q}$. Our main results show $\frac{1}{\delta +1}\le c_{\delta ,q}\le \frac{1}{\lfloor \sqrt{\delta +1}\rfloor +1}$ for all $q\ge 3$ and $\delta \ge 1$, and $c_{\delta ,q}=\frac{1}{\delta +1}$ whenever $\delta \ge 1$ and $q\ge \lceil \frac{3\delta }{2}\rceil +1$.