A Generalization of Nemhauser and Trotter's Local Optimization Theorem

Fellows, Guo, Moser and Niedermeier [JCSS2011] proved a generalization of Nemhauser and Trotter’s theorem, which applies to Bounded-Degree Vertex Deletion (for a fixed integer d ≥ 0, to delete k vertices of the input graph to make the maximum degree of it ≤ d) and gets a linear-vertex kernel for d = 0 and 1, and a superlinear-vertex kernel for each d ≥ 2. It is still left as an open problem whether Bounded-Degree Vertex Deletion admits a linear-vertex kernel for each d ≥ 3. In this paper, we refine the generalized Nemhauser and Trotter’s theorem and get a linear-vertex kernel for each d ≥ 0.