Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions

Abstract

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X subset of 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 a finite integer index and such that YES-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 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-F-DELETION asks whether G has a set X subset of V(G) such that vertical bar X vertical bar <= k and G-X is H-minor-free for every H epsilon F. As our second application, we present the first single-exponential algorithm to solve PLANAR-F-DELETION. Namely, our algorithm runs in time 2())(O(k) . n(2), which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F.