Perfect digraphs

TL;DR

This paper investigates a class of directed graphs called $ ext{α}$-diperfect digraphs, focusing on a conjecture linking their structure to the absence of certain anti-directed odd cycles, and verifies this for specific graph classes.

Contribution

It proves Berge's conjecture for $ ext{α}$-diperfect digraphs within series-parallel and in-semicomplete classes, and proposes a related conjecture verified in known cases.

Findings

Verified Berge's conjecture for series-parallel digraphs

Confirmed the conjecture for in-semicomplete digraphs

Proposed and verified a similar conjecture for known cases

Abstract

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.