Fast FPT-algorithms for cleaning grids

We consider the problem that, given a graph G and a parameter k, asks whether the edit distance of G and a rectangular grid is at most k. We examine the general case where the edit operations are vertex/edge removals and additions. If the dimensions of the grid are given in advance, we give a parameterized algorithm that runs in 2(O(log k.k)) + O(n(3)) steps. In the case where the dimensions of the grid are not given we give a parameterized algorithm that runs in 2(O(log k.k)) + O(k(2.)n(3)) steps. We insist on parameterized algorithms with running times where the relation between the polynomial and the nonpolynomial part is additive. Our algorithm is based on the technique of kernelization. In particular we prove that for each version of the above problem there exists a kernel of size O(k(4)).