Abstract
Let k be a fixed integer. We determine the complexity of finding a p-partition (V1,. .. , Vp) of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by Vi, (1 ≤ i ≤ p) is at least k smaller than the maximum out-degree of D. We show that this problem is polynomial-time solvable when p ≥ 2k and N P-complete otherwise. The result for k = 1 and p = 2 answers a question posed in [3]. We also determine, for all fixed non-negative integers k1, k2, p, the complexity of deciding whether a given digraph of maximum out-degree p has a 2-partition (V1, V2) such that the digraph induced by Vi has maximum out-degree at most ki for i ∈ [2]. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition (V1, V2) such that each vertex v ∈ Vi has at least as many neighbours in the set V3−i as in Vi, for i = 1, 2 is N P-complete. This solves a problem from [6] on majority colourings.