## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Theory of Computing Systems |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - 8 Jul 2015 |

Externally published | Yes |

## Keywords

- Bisection
- Cliquewidth
- NP-hard problems
- Problem kernelization
- Treewidth reduction

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics