The parameterized complexity landscape of finding 2-partitions of digraphs

Abstract

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.