Fixed parameterized algorithms for generalized feedback vertex set problems

A graph is an r-pseudoforest if every connected component of it has a feedback edge set of size at most r. A graph is a d-quasi-forest if every connected component of it has a feedback vertex set of size at most d. The r-Pseudoforest Deletion problem (d-Quasi-Forest Deletion problem) asks to delete a minimum number of vertices to get an r-pseudoforest (a d-quasi-forest, respectively). The well-studied feedback vertex set problem is the special case of r-Pseudoforest Deletion (d-Quasi-Forest Deletion, respectively) in which r=0 (d=0, respectively). We provide an improved FPT algorithm and a smaller kernel for r-Pseudoforest Deletion when parameterized by the solution size (when r is fixed). For d-Quasi-Forest Deletion, we show that it is FPT as well when parameterized by d and the solution size.