Complexity and Kernels for Bipartition into Degree-bounded Induced Graphs

In this paper, we study the parameterized complexity of the problems of partitioning the vertex set of a graph into two parts VA and VB such that VA induces a graph with degree at most a (resp., an a-regular graph) and VB induces a graph with degree at most b (resp., a b-regular graph). These two problems are called Upper-Degree-Bounded Bipartition and Regular Bipartition respectively. First, we prove that the two problems are NP-complete with any nonnegative integers a and b except a = b = 0. Second, we show the fixed-parameter tractability of constrained versions of these two problems with parameter k = |VA| being the size of one part of the bipartition by deriving some problem kernels for them.