### Abstract

Given two disjoint subsets T_{1} and T_{2} of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where |T_{1}| and |T_{2}| are even numbers, we show that V can be partitioned into two sets V_{1} and V_{2} such that the graphs induced by V_{1} and V_{2} are both connected and |V_{1}∩T_{j}| = |V_{2}∩T_{j}| = |T_{j}|/2 holds for each j = 1, 2. Such a partition can be found in O(|V|^{2}log|V|) time. Our proof relies on geometric arguments. We define a new type of 'convex embedding' of k-connected graphs into real space R^{k-1} and prove that for k = 3 such an embedding always exists.

Original language | English |
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Pages (from-to) | 537-554 |

Number of pages | 18 |

Journal | Combinatorica |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2002 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

Nagamochi, H., Jordán, T., Nakao, Y., & Ibaraki, T. (2002). Convex embeddings and bisections of 3-connected graphs.

*Combinatorica*,*22*(4), 537-554. https://doi.org/10.1007/s00493-002-0006-8