Nonconstructive tools for proving polynomial-time decidability
Journal of the ACM (JACM)
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Reduction algorithms for graphs of small treewidth
Information and Computation
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Polynomial-time data reduction for dominating set
Journal of the ACM (JACM)
Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization
Journal of Computer and System Sciences
Improved algorithms for feedback vertex set problems
Journal of Computer and System Sciences
Simpler Parameterized Algorithm for OCT
Combinatorial Algorithms
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Rank-width and tree-width of H-minor-free graphs
European Journal of Combinatorics
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A quartic kernel for pathwidth-one vertex deletion
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Hitting and harvesting pumpkins
ESA'11 Proceedings of the 19th European conference on Algorithms
An O(2O(k)n3) FPT algorithm for the undirected feedback vertex set problem*
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
A single-exponential FPT algorithm for the K4- minor cover problem
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Linear problem kernels for NP-hard problems on planar graphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
Parameterized Complexity
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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 $\mathcal{F}$.