## Abstract

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA'95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.

Original language | English |
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Title of host publication | 14th International Symposium on Parameterized and Exact Computation, IPEC 2019 |

Editors | Bart M. P. Jansen, Jan Arne Telle |

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

ISBN (Electronic) | 9783959771290 |

DOIs | |

State | Published - 1 Dec 2019 |

Event | 14th International Symposium on Parameterized and Exact Computation, IPEC 2019 - Munich, Germany Duration: 11 Sep 2019 → 13 Sep 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 148 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 14th International Symposium on Parameterized and Exact Computation, IPEC 2019 |
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Country/Territory | Germany |

City | Munich |

Period | 11/09/19 → 13/09/19 |

## Keywords

- Carving-width
- Clustered planarity
- FPT
- Non-crossing partitions

## ASJC Scopus subject areas

- Software