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Uniform Kernelization Complexity of Hitting Forbidden Minors

  • Archontia C. Giannopoulou
  • Bart M. P. Jansen
  • Daniel Lokshtanov
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

Abstract

The \(\mathcal {F}\) -Minor-Free Deletion problem asks, for a fixed set \(\mathcal {F}\) and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of \(\mathcal {F} \) as a minor. Fomin et al. (FOCS 2012) showed that the special case when \(\mathcal {F} \) contains at least one planar graph has a kernel of size \(f(\mathcal {F}) \cdot k^{g(\mathcal {F})}\) for some functions f and g. They left open whether this Planar \(\mathcal {F}\) -Minor-Free Deletion problem has kernels whose size is uniformly polynomial, of the form \(f(\mathcal {F}) \cdot k^c\) for some universal constant c. We prove that some Planar \(\mathcal {F}\) -Minor-Free Deletion problems do not have uniformly polynomial kernels (unless NP \(\subseteq \) coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether k vertices can be removed to obtain a graph of treedepth at most \(\eta \). We prove that this problem admits uniformly polynomial kernels with \({\mathcal {O}}(k^6)\) vertices for every fixed \(\eta \).

Keywords

Kernelization Treedepth Minor-free deletion 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Archontia C. Giannopoulou
    • 1
  • Bart M. P. Jansen
    • 2
  • Daniel Lokshtanov
    • 3
  • Saket Saurabh
    • 4
  1. 1.University of WarsawWarsawPoland
  2. 2.Eindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.University of BergenBergenNorway
  4. 4.Institute of Mathematical SciencesChennaiIndia

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