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 .

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [(Meta) kernelization, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2009, pp. 629--638] on graphs of bounded genus, then generalized by Fomin et al. [Bidimensionality and kernels, in Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadephia, 2010, pp. 503--510] to graphs excluding a fixed minor, and by Kim et al. [Linear kernels and single-exponential algorithms via protrusion decompositions, in Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 7965 (2013), pp. 613--624] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions, but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper, we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in counting monadic second order logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for $r$-Dominating Set and $r$-Scattered Set on apex-minor-free graphs, and for Planar-$\mathcal{F}$-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in $\mathcal{F}$ are connected.

The $c$-pumpkin is the graph with two vertices linked by $c \geq 1$ parallel edges. A $c$-pumpkin-model in a graph $G$ is a pair $\{A, B\}$ of disjoint subsets of vertices of $G$, each inducing a connected subgraph of $G$, such that there are at least $c$ edges in $G$ between $A$ and $B$. We focus on hitting and packing $c$-pumpkin-models in a given graph in the realm of approximation algorithms and parameterized algorithms. We give a fixed-parameter tractable (FPT) algorithm running in time $2^{\mathcal{O}(k)} n^{\mathcal{O}(1)}$ deciding, for any fixed $c \geq 1$, whether all $c$-pumpkin-models can be hit by at most $k$ vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases $c=1,2$ respectively. Finally, we present an $\mathcal{O}(\log n)$-approximation algorithm for both the problems of hitting all $c$-pumpkin-models with a smallest number of vertices and packing a maximum number of vertex-disjoint $c$-pumpkin-models.