A cubic kernel for Feedback Vertex Set

In this paper, it is shown that the FEEDBACK VERTEX SET problem on unweighted, undirected graphs has a kernel of cubic size. Le., a polynomial time algorithm is described, that, when given a graph G and an integer k, finds a graph H and integer k' <= k, such that H has a feedback vertex set with at most k' vertices, if and only if G has a feedback vertex set with at most k vertices, and H has at most O(k(3)) vertices and edges. This improves upon a result by Burrage et al. [8] who gave a kernel for FEEDBACK VERTEX SET of size O(k(11)). One can easily make the algorithm constructive, and transform a minimum size feedback vertex set of H with at most, k' vertices into a minimum size feedback vertex set of G. The kernelization algorithm can be used as a first step of an FPT algorithm for FEEDBACK VERTEX SET, but also as a preprocessing heuristic for the problem.