## Abstract

Given a graph G = (V, E) and an integer k, the Cluster Editing problem asks whether we can transform G into a union of vertex-disjoint cliques by at most k modifications (edge deletions or insertions). In this paper, we study the following variant of Cluster Editing. We are given a graph G = (V, E), a packing H of modification-disjoint induced P3s (no pair of P3s in H share an edge or non-edge) and an integer ℓ. The task is to decide whether G can be transformed into a union of vertex-disjoint cliques by at most ℓ + |H| modifications (edge deletions or insertions). We show that this problem is NP-hard even when ℓ = 0 (in which case the problem asks to turn G into a disjoint union of cliques by performing exactly one edge deletion or insertion per element of H) and when each vertex is in at most 23 P3s of the packing. This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by C. Komusiewicz at Shonan meeting no. 144 in March 2019. We then initiate the study to find the largest integer c such that the problem remains tractable when restricting to packings such that each vertex is in at most c packed P3s. Van Bevern et al. showed that the case c=1 is fixed-parameter tractable with respect to ℓ and we show that the case c=2 is solvable in |V |2ℓ+O(1) time.

Original language | English |
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Title of host publication | 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021 |

Editors | Markus Blaser, Benjamin Monmege |

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

ISBN (Electronic) | 9783959771801 |

DOIs | |

State | Published - 1 Mar 2021 |

Externally published | Yes |

Event | 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021 - Virtual, Saarbrucken, Germany Duration: 16 Mar 2021 → 19 Mar 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 187 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021 |
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Country/Territory | Germany |

City | Virtual, Saarbrucken |

Period | 16/03/21 → 19/03/21 |

## Keywords

- Fixed-parameter tractability
- Graph algorithms
- Parameterized complexity

## ASJC Scopus subject areas

- Software

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