## Abstract

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

Original language | English |
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Article number | 35 |

Pages (from-to) | 1-35 |

Journal | ACM Transactions on Algorithms |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Mar 2017 |

## Keywords

- Kernelization
- Lower bounds
- Minor-free deletion
- Treedepth