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

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.