Parameterized algorithms for generalizations of Directed Feedback Vertex Set

The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph G and seeks smallest vertex set S that hits all cycles in . This is one of Karp’s 21 NP -complete problems. Resolving the parameterized complexity status DFVS was long-standing open until Chen et al. (2008) showed its fixed-parameter tractability via 4 k ! n O ( 1 ) -time algorithm, where = | Here we show two generalizations DFVS: • Find such every strong component ? has size at most s : give an algorithm solving this time + ? generalizes by Xiao (2017) for undirected version problem. non-trivial 1-out-regular: 2 3 We also solve corresponding arc versions these problems algorithms.