Let $D$ be a digraph. A stable set $S$ of $D$ and a path partition $\mathcal{P}$ of $D$ are orthogonal if every path $P \in \mathcal{P}$ contains exactly one vertex of $S$. In 1982, Berge defined the class of $\alpha$-diperfect digraphs. A digraph $D$ is $\alpha$-diperfect if for every maximum stable set $S$ of $D$ there is a path partition $\mathcal{P}$ of $D$ orthogonal to $S$ and this property holds for every induced subdigraph of $D$. An anti-directed odd cycle is an orientation of an odd cycle $(x_0,\ldots,x_{2k},x_0)$ with $k\geq2$ in which each vertex $x_0,x_1,\ldots,x_{2k-1}$ is either a source or a sink. Berge conjectured that a digraph $D$ is $\alpha$-diperfect if and only if $D$ does not contain an anti-directed odd cycle as an induced subdigraph. In this paper, we show that this conjecture is false by exhibiting an infinite family of orientations of complements of odd cycles with at least seven vertices that are not $\alpha$-diperfect.

翻译：在1982年, Berge 定义了 $alpha$ - different diperdigraphes 的类别。 如果每条路径 $P\ $\ mathcal{P} $ $ 固定设定为 $S$ 美元, 路径分区为 $\ mathcal{P} $ 美元 如果每条路径 $P\ $\ mathcal{P} 美元每条路径分区为 $\ mathcal{P} 美元是正方形的, 如果每条最稳定的路径分区为 $x_ 0,x_ 1,\ ldots, 美元为 美元, 则路径分区为 $ $ 或 美元 美元, 路径为 美元 。 反方向为每条路径为 $ 美元 。 反方向为 直线的圆周期为 $ 0\, 如果此直径直径直径直径的直径直径直径直径直径直径直径直径直的直径直径直径直径直径直径直径直径直径直径直径直径。