  Let $D = (V(D), A(D))$ be a digraph. A subset $S \subseteq V(D)$ is
$k$-independent if the distance between every pair of vertices of $S$ is at
least $k$, and it is $\ell$-absorbent if for every vertex $u$ in $V(D)
\setminus S$ there exists $v \in S$ such that the distance from $u$ to $v$ is
less than or equal to $\ell$. A $k$-kernel is a $k$-independent and
$(k-1)$-absorbent set. A kernel is simply a $2$-kernel.
  A classical result due to Duchet states that if every directed cycle in a
digraph $D$ has at least one symmetric arc, then $D$ has a kernel. We propose a
conjecture generalizing this result for $k$-kernels and prove it true for $k =
3$ and $k = 4$.
