# Degree constrained 2-partitions of semicomplete digraphs

Jørgen Bang-Jensen*, Tilde My Christiansen

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

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  and  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.

Original language English Theoretical Computer Science 746 112-123 0304-3975 https://doi.org/10.1016/j.tcs.2018.06.028 Published - 25. Oct 2018

### Keywords

• 2-partition
• Digraphs of bounded independence number
• Minimum in-degree
• Minimum out-degree
• Minimum semi-degree
• NP-complete
• Semicomplete digraph
• Tournament