A family of counterexamples for a conjecture of Berge on $\alpha$-diperfect digraphs

Abstract

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 $(x_0,\dots ,x_{2k},x_0)$ with $k\ge 2$ in which each vertex $x_0,x_1,x_2,x_3,x_5,x_7\dots ,x_{2k-1}$ 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.