A Cubic Kernel for Feedback Vertex Set

Abstract

In this paper, it is shown that the Feedback Vertex Set problem on unweighted, undirected graphs has a kernel of cubic size. I.e., 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.