On the parameterized complexity of 2-partitions

We give an FPT algorithm for deciding whether the vertex set of a digraph D can be partitioned into two disjoint sets V 1 , V 2 such that the digraph D [ V 1 ] induced by V 1 has a vertex that can reach all other vertices by directed paths, the digraph D [ V 2 ] has no vertex of in-degree zero and | V i | ≥ k i , where k 1 , k 2 are part of the input. This settles an open problem from [1] , [4] .