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〈Vi〉 has out-degree at least ki 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〈Vi〉 has out-degree at least ki 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〈Vi〉 is a strong tournament.