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 ⊆ 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 Y es -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 , P lanar - F -D eletion asks whether G has a set X ⊆ V ( G ) such that | X | ⩽ k and G − X is H -minor-free for every H ϵ F . As our second application, we present the first single-exponential algorithm to solve P lanar - F -D eletion . 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 .