  Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a
$q$-kernel if $Q$ is an independent set and if every vertex of $D$ can be
reached from $Q$ by a path of length at most $q$. In this paper, we initiate
the study of several extremal problems for $q$-kernels. For example, we
introduce and make progress on (what turns out to be) a weak version of the
Small Quasikernel Conjecture, namely that every digraph contains a $q$-kernel
with $|N^+[Q]|\ge \frac{1}{2}|V(D)|$ for all $q\ge 2$.
