Finding good 2-partitions of digraphs II. Enumerable properties

We continue the study, initiated in [3], of the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties and given minimum cardinality. Let E be the following set of properties of digraphs: E ={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper we determine, for all choices of P1,P2 from E and all pairs of fixed possitive integers k1, k2, the complexity of deciding whether a digraph has a vertex partition into two digraphs D1, D2 such that Di has property Pi and |V (Di)| ≥ ki, i = 1, 2. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the analogous problems when P1 ∈ H and P2 ∈ H ∪ E , where H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in [3]