Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-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. Our results extend the known classes of graphs on which Dominating Set and Connected Dominating Set problems admit linear kernels. Prior to our work, it was known that these problems admit linear kernels on graphs excluding a fixed graph H as a minor. Moreover, for Dominating Set, a kernel of size k, where c(H) is a constant depending on the size of H, follows from a more general result on the kernelization of Dominating Set on graphs of bounded degeneracy. For Connected Dominating Set no polynomial kernel on H-topological-minor free graphs was known prior to our work. On the negative side, it is known that Connected Dominating Set on 2-degenerated graphs does not admit a polynomial kernel unless coNP ⊆ NP/poly. Our kernelization algorithm is based on a non-trivial combination of the following ingredients • The structural theorem of Grohe and Marx [STOC 2012] for graphs excluding a fixed graph H as a topological subgraph; • A novel notion of protrusions, different that the one defined in [FOCS 2009]; • Reinterpretations of reduction techniques developed for kernelization algorithms for Dominating Set and Connected Dominating Set from [SODA 2012]. A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices. Roughly speaking, in the new notion of protrusion instead of demanding the subgraph of being of constant treewidth, we ask it to contain a constant number of vertices from a solution. We believe that the new notion of protrusion will be useful in many other algorithmic settings.