论文标题
针对奇数周期
A half-integral Erdős-Pósa theorem for directed odd cycles
论文作者
论文摘要
我们证明存在一个函数$ f：\ mathbb {n} \ rightArrow \ mathbb {r} $，使每个有向图$ g $都包含$ k $指导的奇数循环，其中每个顶点$ g $都包含在其中大多数$ g $中，或者大多数$ f（k）$ f（k）$ vertices $ vertice colets $ vertices coletles colets cylet cyceles coles cyceles cole cyceles cycecles cole cole cole cole copercy的。我们还为固定$ K $提供了多项式时间算法，该算法输出了两个结果之一。使用此算法结果，我们为固定$ k $提供了多项式时间算法，以决定是否存在这种$ k $指示的奇数循环，或者没有$ k $ tertex-disswischinchechandectectect odecles odcles。这将Reed [Combinatorica 1999]的无向奇数循环扩展到无向奇数的半综合ERDőS-Pósa定理到定向图。
We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$ vertices meeting all directed odd cycles. We also give a polynomial-time algorithm for fixed $k$ which outputs one of the two outcomes. Using this algorithmic result, we give a polynomial-time algorithm for fixed $k$ to decide whether such $k$ directed odd cycles exist, or there are no $k$ vertex-disjoint directed odd cycles. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.
