Some results on Berge's conjecture and Begin-End conjecture

Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $\alpha$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $\alpha$-diperfect if every induced subdigraph of $D$ satisfies the $\alpha$-property. In 1982, Claude Berge proposed a characterization of $\alpha$-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph $D$ satisfies the Begin-End-property or BE-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that (i) $S$ and $\mathcal{P}$ are orthogonal and (ii) for each path $P \in \mathcal{P}$, either the start or the end of $P$ lies in $S$. A digraph $D$ is BE-diperfect if every induced subdigraph of $D$ satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization of BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we show some structural results for $\alpha$-diperfect and BE-diperfect digraphs. In particular, we show that in every minimal counterexample $D$ to both conjectures, the size of a maximum stable set is smaller than $\vert V(D)\vert /2$. As an application we use these results to prove both conjectures for arc-locally in-semicomplete and arc-locally out-semicomplete digraphs.