Some Results on Berge’s Conjecture and Begin–End Conjecture

Abstract

Let D be a digraph. A subset S of V(D) is stable if every pair of vertices in S is non-adjacent in D. A collection of disjoint paths \(\mathcal {P}\) of D is a path partition of V(D), if every vertex in V(D) is on a path of \(\mathcal {P}\). We say that a stable set S and a path partition \(\mathcal {P}\) are orthogonal if every path of \(\mathcal {P}\) contains exactly one vertex of S. A digraph D satisfies the \(\alpha \)-property if for every maximum stable set S of D, there is a path partition \(\mathcal {P}\) orthogonal to S. A digraph D is \(\alpha \)-diperfect if every induced subdigraph of D satisfies the \(\alpha \)-property. In 1982, Berge proposed a characterization of \(\alpha \)-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph D satisfies the Begin–End-property or BE-property if for every maximum stable set S of D, there is a path partition \(\mathcal {P}\) such that (i) S and \(\mathcal {P}\) are orthogonal and (ii) for each path \(P\in \mathcal {P}\), either the initial or the final of P lies in S. A digraph D is BE-diperfect if every induced subdigraph of D satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization of BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we show some structural results for \(\alpha \)-diperfect and BE-diperfect digraphs. In particular, we show that in every minimal counterexample D to both conjectures, it follows that \(\alpha (D) < \vert V(D)\vert /2\). Moreover, we prove both conjectures for arc-locally (out) in-semicomplete digraphs.