  Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path
partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that
$\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S|
= 1$ for every $P \in \mathcal{P}$. We say that $D$ satisfies the
$\alpha$-property if, for every maximum stable set $S$ of $D$, there exists an
$S$-path partition of $D$, and we say that $D$ is $\alpha$-diperfect if every
induced subdigraph of $D$ satisfies the $\alpha$-property. A digraph $C$ is an
anti-directed odd cycle if (i) the underlying graph of $C$ is a cycle $x_1x_2
\cdots x_{2k + 1}x_1$, where $k \in \mathbb{Z}$ and $k \geq 2$, and (ii) each
of the vertices $x_1, x_2, x_3, x_4, x_6,$ $x_8, \ldots, x_{2k}$ is either a
source or a sink. Berge (1982) conjectured that a digraph is $\alpha$-diperfect
if, and only if, it contains no induced anti-directed odd cycle. Remark that
this conjecture is strikingly similar to Berge's conjecture on perfect graphs
-- nowadays known as the Strong Perfect Graph Theorem (Chudnovsky, Robertson,
Seymour, and Thomas, 2006). To the best of our knowledge, Berge's conjecture
for $\alpha$-diperfect digraphs has been verified only for symmetric digraphs
and digraphs whose underlying graph are perfect. In this paper, we verify it
for digraphs whose underlying graphs are series-parallel and for
in-semicomplete digraphs. Moreover, we propose a conjecture similar to Berge's
and verify it for all the known cases of Berge's conjecture.
