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 V-A and V-B such that V-A induces a graph with degree at most a (resp., an a-regular graph) and V-B 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. When a = b = 0, the two problems become the polynomially solvable problem of checking the bipartition of a graph. When a = 0 and b = 1, REGULAR BIPARTITION becomes a well-known NP-hard problem, called DOMINATING INDUCED MATCHING. In this paper, firstly we prove that the two problems are NP-complete with any nonnegative integers a and b except a = b = 0. Secondly, we show the fixed parameter tractability of these two problems with parameter k = |V-A| being the size of one part of the bipartition by deriving several problem kernels for them and constrained versions of them. (C) 2016 Elsevier B.V. All rights reserved.