## Abstract

We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. For a tree T with t nodes, it can be decided in 2
^{O}(
^{t}
^{2 log} t
^{)} · n
^{3} time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

Original language | English |
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Title of host publication | 15th International Symposium on Parameterized and Exact Computation (IPEC 2020) |

Editors | Y. Cao, M. Pilipczuk |

Place of Publication | Dagstuhl, Germany |

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

Pages | 8:1-8:15 |

Volume | 180 |

ISBN (Print) | 978-3-95977-172-6 |

DOIs | |

Publication status | Published - 2020 |

Event | 15th International Symposium on Parameterized and Exact Computation - Online, Hongkong, China Duration: 14 Dec 2020 → 18 Dec 2020 Conference number: 15 https://algo2020.comp.polyu.edu.hk/ipec-cfp.html |

### Publication series

Series | Leibniz International Proceedings in Informatics |
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Volume | 180 |

ISSN | 1868-8969 |

### Symposium

Symposium | 15th International Symposium on Parameterized and Exact Computation |
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Abbreviated title | IPEC 2020 |

Country/Territory | China |

City | Hongkong |

Period | 14/12/20 → 18/12/20 |

Internet address |