### Abstract

Given an undirected and connected graph G=(V,E) and two vertices s,t∈V, a vertex subset S that separates s and t is called an s-t separator, and an s-t separator is called minimal if no proper subset of S separates s and t. Moreover, we say that a set S is a minimal separator of G if S is a minimal s-t separator for some s and t. In this paper, we consider finding a minimal (s-t) separator with maximum weight on a vertex-weighted graph. We first prove that these problems are NP-hard. On the other hand, we give an O⁎(twO(tw))-time deterministic algorithm based on tree decompositions where O⁎ is the order notation omitting the polynomial factor of n. Moreover, we improve the algorithm by using the Rank-Based approach and the running time is O⁎(38⋅2ω)tw. Finally, we give an O⁎(9tw⋅W2)-time randomized algorithm to determine whether there exists a minimal (s-t) separator where W is its weight and tw is the treewidth of G.

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

Pages (from-to) | 294-308 |

Number of pages | 15 |

Journal | Theoretical Computer Science |

Volume | 796 |

DOIs | |

Publication status | Published - 3 Dec 2019 |

### Keywords

- Parameterized algorithm
- Minimal separator
- Treewidth
- TREEWIDTH
- FILL-IN

## Cite this

Hanaka, T., Bodlaender, H. L., van der Zanden, T. C., & Ono, H. (2019). On the maximum weight minimal separator.

*Theoretical Computer Science*,*796*, 294-308. https://doi.org/10.1016/j.tcs.2019.09.025