  Let $D$ be a digraph. We call a subset $N$ of $V(D)$ $k$-independent if for
every pair of vertices $u,v \in N$, $d(u,v) \geq k$; and we call it
$\ell$-absorbent if for every vertex $u \in V(D) \setminus N$, there exists $v
\in N$ such that $d(u,v) \leq \ell$. A $(k,\ell)$-kernel of $D$ is a subset of
vertices which is $k$-independent and $\ell$-absorbent. A $k$-kernel is a
$(k,k-1)$-kernel.
  In this report, we present the main results from our master's research
regarding kernel theory. We prove that if a digraph $D$ is strongly connected
and every cycle $C$ of $D$ satisfies: $(i)$ if $C \equiv 0 \pmod 3$, then $C$
has a short chord and $(ii)$ if $C \not \equiv 0 \pmod 3$, then $C$ has three
short chords: two consecutive and a third crossing one of the former, then $D$
has a $3$-kernel. Moreover, we introduce a modification of the substitution
method, proposed by Meyniel and Duchet in 1983, for $3$-kernels and use it to
prove that a quasi-$3$-kernel-perfect digraph $D$ is $3$-kernel-perfect if
every circuit of length not dividable by three has four short chords.
