The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph $G$ and seeks a smallest vertex set $S$ that hits all cycles in $G$. This is one of Karp’s 21 $\mathsf{NP}$-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a $4^{k}k!n^{\mathcal{O}\left(1\right)}$-time algorithm, where $k=\left|S\right|$.

Here we show fixed-parameter tractability of two generalizations of DFVS:

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Find a smallest vertex set $S$ such that every strong component of $G-S$ has size at most $s$: we give an algorithm solving this problem in time $4^k(ks+k+s)!\cdot n^{\mathcal{O}\left(1\right)}$. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.

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Find a smallest vertex set $S$ such that every non-trivial strong component of $G-S$ is 1-out-regular: we give an algorithm solving this problem in time $2^{\mathcal{O}\left(k^3\right)}\cdot n^{\mathcal{O}\left(1\right)}$.

有向反馈顶点集(DFVS) 问题将有向图作为输入 $G$并寻找一个最小的顶点集 $\mathrm{小号}$击中所有周期$G$. 这是卡普的 21 个之一$\mathsf{NP}$- 完整的问题。在 Chen 等人之前，解决 DFVS 的参数化复杂性状态是一个长期悬而未决的问题。（2008 年）通过$4^{ķ}ķ！n^{\mathcal{○}\left(1\right)}$-时间算法，其中$ķ=\left|\mathrm{小号}\right|$.

在这里，我们展示了 DFVS 的两个推广的固定参数可处理性：

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找到一个最小的顶点集$\mathrm{小号}$使得每一个强大的组成部分$G-\mathrm{小号}$最多有大小 $s$: 我们给出一个算法及时解决这个问题$4^{ķ}(ķs+ķ+s)！\cdot n^{\mathcal{○}\left(1\right)}$. 这概括了 Xiao (2017) 针对该问题的无向版本提出的算法。

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找到一个最小的顶点集$\mathrm{小号}$这样每个非平凡的强成分$G-\mathrm{小号}$is 1-out-regular: 我们给出一个算法及时解决这个问题$2^{\mathcal{○}\left({ķ}^3\right)}\cdot n^{\mathcal{○}\left(1\right)}$.