On the existence of $3$- and $4$-kernels in digraphs

Abstract: 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 $k = 4$.