Polynomial Algorithms for Minimum Degree Partitions in Semicomplete Digraphs

A 2-partition of a digraph is a partition of its vertex set into two nonempty parts. Degree-constrained 2-partition problems are generally computationally difficult, even when the prescribed properties are expressed only in terms of minimum indegree, minimum outdegree, or minimum semidegree. Bang-Jensen and Christiansen~\cite{B-C} conjectured that the minimum-degree partition problems would be polynomial-time solvable on semicomplete digraphs when the degree thresholds are fixed, and Bang-Jensen and Gutin~\cite{B-G-Classes} posed the related Problems~2.8.15 and~2.8.16.
We resolve this conjecture. More precisely, for every fixed pair of integers $k_1,k_2\ge 2$, we give deterministic polynomial-time algorithms that decide whether a given semicomplete digraph admits a $(\delta^+\geq k_1,\delta^-\geq k_2)$-partition, a $(\delta^+\geq k_1,\delta^0\geq k_2)$-partition, or a $(\delta^0\geq k_1,\delta^0\geq k_2)$-partition, and construct such a partition whenever one exists. Here, $\delta ^+,\delta ^-,\delta ^0$ represent the minimum out-, in-, semi-degree, respectively. The algorithms use small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring, which develop a new method in partition algorithm construction.