For a > 1, an a-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I, k) (of a problem Q) of size bounded by a function of k such that, for every c = 1, a c-approximate solution for the new instance can be turned into a (c · a)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every a > 1, Connected Dominating Set admits a polynomial-size a-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP ? coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r ? N there cannot exist an a-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any a > 1 (assuming the Gap Exponential Time Hypothesis). © Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz.