## Abstract

Given a graph G and a parameter k, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset U ⊆ V(G) of size at most k that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for 1 CVD of size O(k ^{161} log ^{58} k) and asked whether one can design a kernel of size O(k ^{10} ) [Jansen an Pilipczuk, SODA 2017]. While we do not completely resolve this question, we design a significantly smaller kernel of size O(k ^{12} log ^{10} k), inspired by the O(k ^{2} )-size kernel for Feedback Vertex Set [Thomassé, TALG 2010]. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.

Original language | English |
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Article number | a11 |

Journal | ACM Transactions on Algorithms |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 1 Dec 2018 |

## Keywords

- Chordal graph
- Chordal vertex deletion
- Kernelization
- Parameterized complexity

## ASJC Scopus subject areas

- Mathematics (miscellaneous)