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 $x_0,x_1,x_2,x_3,x_5,x_7\ldots,x_{2k-1}$ 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。 如果每个最稳定设定的$S$, 一个固定的美元和路径分区 $mathcal{P} $D$ 如果每个路径 $P\ in\ mathcal{P}$ 美元包含一个精确的 $S$的顶点。 1982年, Berge 定义了$\ alpha$ - different diperdigraphy 的类别。 如果每个顶点 $x_0,x_1,x_2,x_3,x_5,x_7>lockaldal=$美元, 美元为美元, 而这个属性是每个引出子目录 $D$的顶点。 反方向是 $xxxxxxxxxxxxxxxxxxxxxxxxxxxxx_x_x_x_x_5, 美元xxxxx_dolgardalal $_lxxx_lx_lx_lxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx