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∣≥ki, where k1,k2 are part of the input. This settles an open problem from[1,4].
@article{arxiv.2003.07190,
title = {On the parameterized complexity of 2-partitions},
author = {Jonas Bamse Andersen and Jørgen Bang-Jensen and Anders Yeo},
journal= {arXiv preprint arXiv:2003.07190},
year = {2020}
}