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 EE be the following set of properties of digraphs: E=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,P2P1,P2 from EE and all pairs of fixed positive integers k1,k2k1,k2, the complexity of deciding whether a digraph has a vertex partition into two digraphs D1,D2D1,D2 such that DiDi has property PiPi and |V(Di)|≥ki|V(Di)|≥ki, i=1,2i=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∈HP1∈H and P2∈H∪EP2∈H∪E, where H=H={acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in [3].