On a generalization of Nemhauser and Trotter's local optimization theorem

Fellows, Guo, Moser and Niedermeier [14] proved a generalization of Nemhauser and Trotter's theorem, which applies to d-Bounded-Degree Vertex Deletion (to delete k vertices of the input graph to make the maximum degree of it ≤d) and gives a linear-vertex kernel for d=0 and 1, and a superlinear-vertex kernel for each d≥2 for the problem parameterized by k. It is still left as an open problem whether this parameterized problem admits a linear-vertex kernel for each d≥3. In this paper, we refine the generalized Nemhauser and Trotter's theorem and give a linear-vertex kernel for each d≥0.