Degree constrained 2-partitions of semicomplete digraphs

A 2-partition of a digraph D is a partition (V₁,V₂) of V(D) into two disjoint non-empty sets V₁ and V₂ such that V₁∪V₂=V(D). A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following: • For all integers k₁,k₂≥1 and k₁+k₂≥3 it is NP-complete to decide whether a given digraph D has a 2-partition (V₁,V₂) such that D〈V_i〉 has out-degree at least k_i for i=1,2.• For every fixed choice of integers α,k₁,k₂≥1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition (V₁,V₂) such that D〈V_i〉 has out-degree at least k_i for i=1,2.• For every fixed integer k≥1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition (V₁,V₂) such that D〈V₁〉 has out-degree at least one and D〈V₂〉 has in-degree at least k.• It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition (V₁,V₂) such that D〈V_i〉 is a strong tournament.