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 P ₁,P ₂ from E and all pairs of fixed positive integers k ₁,k ₂, the complexity of deciding whether a digraph has a vertex partition into two digraphs D ₁,D ₂ such that D _i has property P _i and |V(D _i)|≥k _i, 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 P ₁∈H and P ₂∈H∪E, where H={acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in [3].