On the parameterized complexity of 2-partitions

We give an FPT algorithm for deciding whether the vertex set a digraph $D$ can be partitioned into two disjoint sets $V1,V2$ such that the digraph $D[V1]$ induced by $V1$ has a vertex that can reach all other vertices by directed paths, the digraph $D[V2]$ has no vertex of in-degree zero and $|Vi|\geq ki$, where $k1,k_2$ are part of the input. This settles an open problem from[1,4].