  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.
