Out-degree reducing partitions of digraphs

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V1, \dots, Vp)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $Vi$, ($1\leq i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \cite{bangTCS636}. 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 $i\in [2]$. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition $(V1,V2)$ such that each vertex $v\in Vi$ has at least as many neighbours in the set $V{3-i}$ as in $Vi$, for $i=1,2$ is ${\cal NP}$-complete. This solves a problem from \cite{kreutzerEJC24} on majority colourings.