The parameterized complexity landscape of finding 2-partitions of digraphs

Given a network modeled by a directed graph D=(V,A), it is natural to ask whether we can partition the vertex set of D into two disjoint subsets V₁,V₂ (called a 2-partition), such that the digraphs D[V₁],D[V₂] induced by each of these has one of the two properties of interest. This question gives rise to a rich realm of combinatorial problems. The complexity of many such problems was determined in [2,3]. We analyze a subset of those problems from the viewpoint of parameterized complexity, and present a complete dichotomy of basic, natural properties. More precisely, given a directed graph D=(V,A) and two non-negative integers k₁ and k₂, we seek a 2-partition (V₁,V₂) of the vertex set V such that |V₁|≥k₁, |V₂|≥k₂, and each of the subdigraphs induced by V₁ and V₂ has a structural property as defined by the problem at hand—for example, D[V₁] is acyclic and D[V₂] is strongly connected. Specifically, we consider the following eight structural properties: being strongly connected; being connected; having an out-branching; having an in-branching; having minimum degree at least one; having minimum semi-degree at least one; being acyclic; and being complete.