Linear Kernels and Single-Exponential Algorithms via Protrusion Decompositions

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X⊆V(G), called a treewidth-modulator, such that the treewidth of G−X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs. Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar$\mathcal{F}$- Deletion asks whether G has a set X⊆V(G) such that $|X|\leqslant k$ and G−X is H-minor-free for every $H\in \mathcal{F}$. As our second application, we present the first single-exponential algorithm to solve Planar$\mathcal{F}$- Deletion. Namely, our algorithm runs in time 2O(k)·n2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family $\mathcal{F}$.