## Abstract

The notion of H-treewidth, where H is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of H-treewidth at most k can be decomposed into (arbitrarily large) H-subgraphs which interact only through vertex sets of size O(k) which can be organized in a tree-like fashion. H-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for H-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of H. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree H-decompositions. We present FPT-approximation algorithms to compute tree H-decompositions for hereditary and union-closed graph classes H. Given a graph of H-treewidth k, we can compute a 5-approximate tree H-decomposition in time f(O(k)) · n^{O}^{(1)} whenever H-deletion parameterized by solution size can be solved in time f(k)·n^{O}^{(1)} for some function f(k) ≥ 2^{k}. The current-best algorithms either achieve an approximation factor of k^{O}^{(1)} or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^{O}(k^{)} · n^{O}^{(1)} parameterized by bipartite-treewidth and Vertex Planarization in time 2^{O}(k log k^{)} · n^{O}^{(1)} parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.

Original language | English |
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Title of host publication | 31st Annual European Symposium on Algorithms, ESA 2023 |

Editors | Inge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772952 |

DOIs | |

State | Published - 1 Sep 2023 |

Externally published | Yes |

Event | 31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands Duration: 4 Sep 2023 → 6 Sep 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 274 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Annual European Symposium on Algorithms, ESA 2023 |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 4/09/23 → 6/09/23 |

## Keywords

- fixed-parameter tractability
- graph decompositions
- treewidth

## ASJC Scopus subject areas

- Software