## Abstract

Vertex-deletion problems have been at the heart of parameterized complexity throughout its history. Here, the aim is to determine the minimum size (denoted by modH) of a modulator to a graph class H, i.e., a set of vertices whose deletion results in a graph in H. Recent years have seen the development of a research programme where the complexity of modulators is measured in ways other than size. For instance, for a graph class H, the graph parameters elimination distance to H (denoted by edH) [Bulian and Dawar, Algorithmica, 2016] and H-treewidth (denoted by twH) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the “torso” of the graph induced on a modulator to the graph class H. Here, the torso of a vertex set S in a graph G is the graph with vertex set S and an edge between two vertices u, v ∈ S if there is a path between u and v in G whose internal vertices all lie outside S. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class H that satisfies mild additional conditions. In fact, we show that for every hereditary graph class H satisfying mild

additional conditions, with the exception of edH parameterized by twH, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to H) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing edH and twH.

The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if H is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a “strong irrelevant vertex rule”, then there exists a uniform FPT algorithm for edH. Using these sufficient conditions, we obtain uniform FPT algorithms for computing edH, when H is defined by excluding a finite number of connected (a) minors, or (b) topological minors, or (c) induced subgraphs, or when H is any of bipartite, chordal or interval graphs. For most of these problems, the existence of a uniform FPT algorithm has remained open in the literature. In fact, for some of them, even a non-uniform FPT algorithm was not known. For example, Jansen et al. [STOC 2021] ask for such an algorithm when H is defined by excluding a finite number of connected topological minors. We resolve their question in the affirmative.

additional conditions, with the exception of edH parameterized by twH, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to H) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing edH and twH.

The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if H is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a “strong irrelevant vertex rule”, then there exists a uniform FPT algorithm for edH. Using these sufficient conditions, we obtain uniform FPT algorithms for computing edH, when H is defined by excluding a finite number of connected (a) minors, or (b) topological minors, or (c) induced subgraphs, or when H is any of bipartite, chordal or interval graphs. For most of these problems, the existence of a uniform FPT algorithm has remained open in the literature. In fact, for some of them, even a non-uniform FPT algorithm was not known. For example, Jansen et al. [STOC 2021] ask for such an algorithm when H is defined by excluding a finite number of connected topological minors. We resolve their question in the affirmative.

Original language | English |
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Title of host publication | ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |

Publisher | Association for Computing Machinery |

Pages | 1976-2004 |

Number of pages | 29 |

ISBN (Electronic) | 9781611977073 |

DOIs | |

State | Published - 2022 |

Event | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States Duration: 9 Jan 2022 → 12 Jan 2022 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2022-January |

### Conference

Conference | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |
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Country/Territory | United States |

City | Alexander |

Period | 9/01/22 → 12/01/22 |

## Keywords

- Computer Science - Data Structures and Algorithms

## ASJC Scopus subject areas

- Software
- Mathematics (all)