Linear kernels for separating a graph into components of bounded size

The p-Size Separator problem is to check whether we can delete at most k vertices in a given graph such that each connected component of the remaining graph has at most p vertices. We prove a kernel of O(pk) vertices for this problem, i.e., a linear vertex kernel for each fixed p≥1. Our proofs are based on extremal combinatorial arguments and the main result can be regarded as a generalization of the Nemhauser and Trotter's theorem for the Vertex Cover problem. Our techniques are possible to be used to improve kernel algorithms based on the expansion lemma or crown decompositions.