Let D be a digraph. A subset S of V (D) is a stable set if every pair of vertices in S is non-adjacent in D. A collection of disjoint paths is a path partition of D, if every vertex in V (D) is in exactly one path of ...Let D be a digraph. A subset S of V (D) is a stable set if every pair of vertices in S is non-adjacent in D. A collection of disjoint paths is a path partition of D, if every vertex in V (D) is in exactly one path of . We say that a stable set S and a path partition are orthogonal if each path of contains exactly one vertex of S. A digraph D satisfies the α-property if for every maximum stable set S of D, there exists a path partition such that S and are orthogonal. A digraph D is α-diperfect if every induced subdigraph of D satisfies the α-property. In 1982, Berge proposed a characterization for α-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 exists a path partition such that 1) S and are orthogonal and 2) for each path P ∈ , either the start or the end of P belongs to S. A digraph D is BE-diperfect if every induced subdigraph of D satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for 3-anti-circulant digraphs. We also present some structural results for α-diperfect and BE-diperfect digraphs.展开更多
文摘Let D be a digraph. A subset S of V (D) is a stable set if every pair of vertices in S is non-adjacent in D. A collection of disjoint paths is a path partition of D, if every vertex in V (D) is in exactly one path of . We say that a stable set S and a path partition are orthogonal if each path of contains exactly one vertex of S. A digraph D satisfies the α-property if for every maximum stable set S of D, there exists a path partition such that S and are orthogonal. A digraph D is α-diperfect if every induced subdigraph of D satisfies the α-property. In 1982, Berge proposed a characterization for α-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 exists a path partition such that 1) S and are orthogonal and 2) for each path P ∈ , either the start or the end of P belongs to S. A digraph D is BE-diperfect if every induced subdigraph of D satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for 3-anti-circulant digraphs. We also present some structural results for α-diperfect and BE-diperfect digraphs.
