Kernels for (Connected) Dominating Set on Graphs with Excluded Topological Minors

Abstract

We give the first linear kernels for the D ominating S et and C onnected D ominating S et problems on graphs excluding a fixed graph H as a topological minor. In other words, we prove the existence of 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 one. Our results extend the known classes of graphs on which the D ominating S et and C onnected D ominating S et problems admit linear kernels. Prior to our work, it was known that these problems admit linear kernels on graphs excluding a fixed apex graph H as a minor. Moreover, for D ominating S et , a kernel of size k c ( H ) , where c ( H ) is a constant depending on the size of H , follows from a more general result on the kernelization of D ominating S et on graphs of bounded degeneracy. Alon and Gutner explicitly asked whether one can obtain a linear kernel for D ominating S et on H -minor-free  graphs. We answer this question in the affirmative and in fact prove a more general result. For C onnected D ominating S et no polynomial kernel even on H -minor-free  graphs was known prior to our work. On the negative side, it is known that C onnected D ominating S et 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 minor; • A novel notion of protrusions, different than the one defined in [FOCS 2009]; • Our results are based on a generic reduction rule that produces an equivalent instance (in case the input graph is H -minor-free) of the problem, with treewidth O (√ k ). The application of this rule in a divide-and-conquer fashion, together with the new notion of protrusions, gives us the linear kernels. A protrusion in a graph [FOCS 2009] is a subgraph of constant treewidth which is separated from the rest of the graph by at most a constant number of vertices. In our variant of protrusions, instead of stipulating that the subgraph be of constant treewidth , we ask that it contains a constant number of vertices from a solution . We believe that this new take on protrusions would be useful for other graph problems and in different algorithmic settings.