Hitting forbidden subgraphs in graphs of bounded treewidth

@inproceedings{Cygan2014HittingFS,
  title={Hitting forbidden subgraphs in graphs of bounded treewidth},
  author={Marek Cygan and D{\'a}niel Marx and Marcin Pilipczuk and Michal Pilipczuk},
  booktitle={Inf. Comput.},
  year={2014}
}

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References

SHOWING 1-10 OF 31 REFERENCES

Easy Problems for Tree-Decomposable Graphs

On Bounded-Degree Vertex Deletion parameterized by treewidth

A linear time algorithm for finding tree-decompositions of small treewidth

TLDR
Every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm that determines whether the treewidth of G is at most k, and if so, finds a treedecomposition of G withtreewidth at mostK.

Incompressibility through Colors and IDs

TLDR
This paper shows how to combine results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems, and rules out the existence of compression algorithms for many of the problems in question.

A Near-Optimal Planarization Algorithm

TLDR
A dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2O(w log w) · n, thereby improving over previous double-exponential algorithms.

Known algorithms on graphs of bounded treewidth are probably optimal

TLDR
Lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth are obtained and the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi are proved.

Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms

TLDR
Two algorithms computing representative families of linear and uniform matroids are given and how to use representative families for designing single-exponential parameterized and exact exponential time algorithms are demonstrated.

An O(c^k n) 5-Approximation Algorithm for Treewidth

TLDR
This is the first algorithm providing a constant factor approximation for tree width which runs in time single-exponential in k and linear in n, and can be used to speed up many such algorithms to work in time which is single-Exponential in the tree width andlinear in the input size.

Treewidth, Computations and Approximations

  • T. Kloks
  • Mathematics
    Lecture Notes in Computer Science
  • 1994
TLDR
Testing superperfection of k-trees and triangulating 3-colored graphs results in approximating treewidth and pathwidth for some classes of perfect graphs.