Linear kernels for (connected) dominating set on H-minor-free graphs

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if and only if G′ has. Prior to our work, the only polynomial kernel for Dominating Set on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is k, where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for Dominating Set on H-minor free graphs. We answer this question in affirmative. For Connected Dominating Set no polynomial kernel on H-minor free graphs was known prior to our work. Our results are based on a novel generic reduction rule producing an equivalent instance of the problem with treewidth O( √ k). The application of this rule in a divide-and-conquer fashion together with protrusion techniques brings us to linear kernels. As a byproduct of our results we obtain the first subexponential time algorithms for Connected Dominating Set, a deterministic algorithm solving the problem on an n-vertex H-minor free graph in time 2O( √ k log k) + nO(1) and a Monte Carlo algorithm of running time 2O( √ k) + nO(1). For Dominating Set our results implies a significant simplification and refinement of a 2O( √ k)nO(1) algorithm on H minor free graphs due to Demaine et al. [SODA 2003, J. ACM 2005]. ∗Department of Informatics, University of Bergen, Norway. Supported by the European Research Council (ERC) advanced grant PREPROCESSING, reference 267959. †University of California, San Diego, USA. ‡The Institute of Mathematical Sciences, CIT Campus, Chennai, India. §Department of Mathematics, National & Kapodistrian University of Athens, Greece. Supported by the project “Kapodistrias” (AΠ 02839/28.07.2008) of the National and Kapodistrian University of Athens (project code: 70/4/8757).