3-反循环有向图是α-双光子和BE双光子

Lucas Freitas, Orlando Lee
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引用次数: 2

摘要

让$D$是一个有向图。如果$S$中的每对顶点在$D$中是不相邻的,则$V(D)$的子集$S$是稳定集。$D$的不相交路径$\mathcal{P}$的集合是$V(D)$的路径分区,如果$V(D)$中的每个顶点都恰好在$\mathical{P}$的路径上。我们说稳定集$S$和路径分区$\mathcal{P}$是正交的,如果$P$的每条路径恰好包含$S$的一个顶点。有向图$D$满足$\alpha$-性质,如果对于$D$的每个最大稳定集$S$,存在一个路径分区$\mathcal{P}$,使得$S$和$\mathcal{P}$正交。如果$D$的每个诱导子图都满足$\alpha$-性质,则有向图$D$是$\alpa$-完全有向图。1982年,Claude Berge提出了$\alpha$-二完全有向图的一个特征化,用禁止的反定向奇环来表示。2018年,桑比内利、席尔瓦和李提出了类似的猜想。有向图$D$满足Begin-End性质或BE性质,如果对于$D$的每个最大稳定集$S$,存在一个路径分区$\mathcal{P}$,使得(i)$S$和$\mathical{P}$正交,并且(ii)对于每个路径$P\in\mathcal{P}$,$P$的开始或结束都属于$S$。如果$D$的每个诱导子图都满足BE性质,则有向图$D$是BE完全图。Sambinelli、Silva和Lee提出了用禁止阻塞奇环刻画BE完全有向图的一个性质。在本文中,我们验证了$3$-反循环有向图的两个猜想。我们还给出了$\alpha$-双完全和BE双完全有向图的一些结构结果。
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3-Anti-Circulant Digraphs Are α-Diperfect and BE-Diperfect
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 $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $\alpha$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $\alpha$-diperfect if every induced subdigraph of $D$ satisfies the $\alpha$-property. In 1982, Claude Berge proposed a characterization for $\alpha$-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 exists 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 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 $\alpha$-diperfect and BE-diperfect digraphs.
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