We prove that given an undirected graph G on n vertices and an integer k, one can compute, in polynomial time in n, a graph G with at most 4k 2 vertices and an integer k such that G has a feedback vertex set of size at most k iff G has a feedback vertex set of size at most k . This result improves a previous O(k 11 ) kernel of Burrage et al. [5] , and a more recent cubic kernel of Bodlaender [3] . This problem was communicated by Fellows in [4] .