The feedback vertex set problem is one of the most studied parameterized problems. Several generalizations of the problem have been studied where one is to delete vertices to obtain graphs close to acyclic. In this paper, we give an FPT algorithm for the problem of deleting at most $k$ vertices to get an $r$-pseudoforest. A graph is an $r$-pseudoforest if we can delete at most $r$ edges from each component to get a forest. Philip et al. introduced this problem and gave an $O^*(c_{r}^{k})$ algorithm for it, where $c_r$ depends on $r$ double exponentially. In comparison, our algorithm runs in time $O^*((10k)^{k})$, independent of $r$.

中文翻译：

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独立于r的时间解决r-pseudoforest删除问题

反馈顶点集问题是研究最多的参数化问题之一。已经研究了该问题的几种概括，其中一种是删除顶点以获得接近非循环的图。在本文中，我们给出了一个FPT算法，用于解决最多删除$ k $个顶点以获得$ r $ -pseudoforest的问题。如果我们可以从每个组件中最多删除$ r $个边以获得一个森林，则图就是$ r $ -pseudoforest。Philip等。引入了这个问题，并给出了一个$ O ^ *（c_ {r} ^ {k}）$算法，其中$ c_r $依赖于$ r $指数倍增长。相比之下，我们的算法在时间$ O ^ *（（10k）^ {k}）$中运行，独立于$ r $。