A Cubic Kernel for Feedback Vertex Set

The FEEDBACK VERTEX SET problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. [7], we show that this problem has a kernel with O(κ3) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer κ, finds a graph G' and integer κ', such that G has a feedback vertex set of size at most k, if and only if G' has a feedback vertex set of size at most k', and G' has O(κ3) vertices. Moreover, if G' has a feedback vertex set of size at most κ', then a minimum size feedback vertex set of G' directly gives a minimum size feedback vertex set of size κ'. This kernelization algorithm can be used as the first step of an FPT algorithm for FEEDBACK VERTEX SET, but also as a preprocessing heuristic for FEEDBACK VERTEX SET