Two edge modification problems without polynomial kernels

Given a graph G and an integer k, the ? Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property ?. Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tractable and some are known to admit polynomial kernelizations, i.e., efficient preprocessing with a size guarantee that is polynomial in k. This paper answers an open problem posed by Cai (IWPEC 2006), namely, whether the ? Edge Deletion problem, parameterized by the number of deletions, admits a polynomial kernelization when ? can be characterized by a finite set of forbidden induced subgraphs. We answer this question negatively based on recent work by Bodlaender et al. (ICALP 2008) which provided a framework for proving polynomial lower bounds for kernelizability. We present a graph H on seven vertices such that $\mathcal{H}$-free Edge Deletion and H-free Edge Editing do not admit polynomial kernelizations, unless $\mbox{NP}\subseteq \mbox{coNP}/\mbox{poly}$. The application of the framework is not immediate and requires a lower bound for a Not-1-in-3 SAT problem that may be of independent interest.