### Аннотация

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the VertexBisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that VertexBisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n^{O(d)}. We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.

Язык оригинала | английский |
---|---|

Журнал | Theory of Computing Systems |

Том | 57 |

Номер выпуска | 1 |

DOI | |

Состояние | Опубликовано - 8 июл 2015 |

Опубликовано для внешнего пользования | Да |

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## Цитировать

*Theory of Computing Systems*,

*57*(1). https://doi.org/10.1007/s00224-014-9557-5