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Let $D$ be a digraph. We call a subset $N$ of $V(D)$ $k$-independent if forevery 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 ofvertices 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 researchregarding kernel theory. We prove that if a digraph $D$ is strongly connectedand 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 threeshort chords: two consecutive and a third crossing one of the former, then $D$has a $3$-kernel. Moreover, we introduce a modification of the substitutionmethod, proposed by Meyniel and Duchet in 1983, for $3$-kernels and use it toprove that a quasi-$3$-kernel-perfect digraph $D$ is $3$-kernel-perfect ifevery circuit of length not dividable by three has four short chords.

Two novel results on the existence of $3$-kernels in digraphs