论文标题
一个对berge的概述的家族，以$α$ -Diperfect Digraphs
A family of counterexamples for a conjecture of Berge on $α$-diperfect digraphs
论文作者
论文摘要
令$ d $为挖掘物。如果每个路径$ p \ in \ Mathcal {p} $中的每个路径$ p \，则稳定的$ s $ $ d $和路径分区$ \ MATHCAL {p} $都是正交的。 1982年，伯格（Berge）定义了$α$ diperfect digraphs的类别。 Digraph $ d $是$α$ -DiperFect，如果对于每个最大稳定套件的$ s $ $ d $，则有一个路径分区$ \ MATHCAL {p} $ $ d $ d $ orthoconal到$ s $，并且此属性均具有$ d $的每个引起的subsDigraph。反向指导的奇数周期是一个奇数周期$（x_0，\ ldots，x_ {2k}，x_0）$的方向，其中每个顶点$ k \ geq2 $中，每个顶点$ x_0，x_1，x_1，x_2，x_2，x_3，x_3，x_3，x_5，x_5，x_7，x_7 \ ldots，x_7 \ ldots，x_____________ {2K-1}伯格（Berge）猜想，当$ d $ d $不包含反向指导的奇数周期作为诱导子数字时，digraph $ d $是$α$ - 二级联。在本文中，我们通过表现出至少七个不是$α$ diperfect的顶点的奇数循环的无限定位来表明这种猜想是错误的。
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 $α$-diperfect digraphs. A digraph $D$ is $α$-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 $α$-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 $α$-diperfect.
