A family of counterexamples for a conjecture of Berge on α-diperfect digraphs

Silva, Caroline Aparecida de Paula; Silva, Cândida Nunes da; Lee, Orlando

doi:10.1016/j.disc.2023.113458

Cited by 1 publication

(3 citation statements)

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“…All the counterexamples of Conjecture 5 given by de Paula Silva, Nunes da Silva and Lee [10] have stability number two and are important examples that illustrate Corollary 11. So far, there is no characterization of α-diperfect digraphs with stability number two.…”

Section: Arborescences In Digraphsmentioning

confidence: 89%

“…In contrast, Berge's original conjecture regarding α-diperfect digraphs (Conjecture 5) is false for digraphs with stability number two. As we mentioned in the introduction, de Paula Silva et al [10] exhibited orientations of complements of odd cycles with at least seven vertices which are not α-diperfect. So anti-directed odd cycles and those orientations of complement of odd cycles are all minimal non-α-diperfect digraphs.…”

Section: Final Remarksmentioning

confidence: 94%

“…In 2023, de Paula Silva, Nunes da Silva and Lee [10] showed that Berge's Conjecture is false by presenting an infinite family of non-α-diperfect super-orientations of complements of odd cycles with at least seven vertices. On the other hand, these digraphs are not counterexamples to Conjecture 6.…”

Section: Introductionmentioning

confidence: 99%

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BE-Diperfect Digraphs with Stability Number Two

A. de Paula Silva,

Nunes da Silva,

Lee

2024

Electron. J. Combin.

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In 1982, Berge defined the class of $\alpha$-diperfect digraphs. A digraph $D$ is $\alpha$-diperfect if every induced subdigraph of $H$ of $D$ satisfies the following property: for every maximum stable set $S$ of $H$ there is a path partition $\mathcal{P}$ of $H$ in which every $P \in \mathcal{P}$ contains exactly one vertex of $S$. Berge conjectured a characterization of $\alpha$-diperfect digraphs by forbidding induced orientations of odd cycles. In 2018, Sambinelli, Nunes da Silva and Lee proposed a similar class of digraphs. A digraph $D$ is BE-diperfect if every induced subdigraph $H$ of $D$ satisfies the following property: for every maximum stable set $S$ of $H$ there is a path partition $\mathcal{P}$ of $H$ in which (i) every $P \in \mathcal{P}$ contains exactly one vertex of $S$ and (ii) $P$ either begins or ends at a vertex of $S$. They also conjectured that the BE-diperfect digraphs can be characterized by forbidding induced orientations of odd cycles; we refer to this as the Begin-End Conjecture. In 2023, de Paula Silva, Nunes da Silva and Lee presented an infinite family of counterexamples with stability number two to Berge's Conjecture. On the other hand, these digraphs are not counterexamples to the Begin-End Conjecture. In this paper, we prove that the latter conjecture holds for digraphs with stability number two.

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Section: Arborescences In Digraphsmentioning

confidence: 89%

Section: Final Remarksmentioning

confidence: 94%