The exact asymptotic bound conjecture for small q-kernels

For integers $q\geq 2$ and $\delta\geq 1$, let $c_{\delta,q}$ be the smallest constant such that every finite digraph $D$ with minimum in-degree at least $\delta$ contains a $q$-kernel of size at most $c_{\delta,q}|V(D)|$. A $q$-kernel is an independent set $Q\subseteq V(D)$ from which every vertex is reachable by a directed path of length at most $q$.

Exact asymptotic bound conjecture. For all $\delta\geq 1$ and $q\geq 3$,

$$c_{\delta,q}=\frac{1}{\delta+1}.$$

The paper establishes the lower bound $1/(\delta+1)\leq c_{\delta,q}$ and an upper bound of $1/(\lfloor\sqrt{\delta+1}\rfloor+1)$ for $q\geq 3$, while equality is proved when $q\geq \lceil 3\delta/2\rceil+1$. The conjecture remains unresolved in the remaining parameter ranges.