FPT algorithms for generalized feedback vertex set problems

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An r-pseudoforest is a graph in which each component can be made into a forest by deleting at most r edges, and a d-quasi-forest is a graph in which each component can be made into a forest by deleting at most d vertices. In this paper, we study the parameterized tractability of deleting minimum number of vertices to obtain r-pseudoforest and d-quasi-forest, generalizing the well studied feedback vertex set problem. We first provide improved FPT algorithm and kernelization results for the r-pseudoforest deletion problem and then we show that the d-quasi-forest deletion problem is also FPT.