On the parameterized complexity of 2-partitions

Abstract: We give an FPT algorithm for deciding whether the vertex set 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 $|V_i|\geq k_i$, where $k_1,k_2$ are part of the input. This settles an open problem from[1,4].