### 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 - 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

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

**Convex embeddings and bisections of 3-connected graphs.** / Nagamochi, Hiroshi; Jordán, T.; Nakao, Yoshitaka; Ibaraki, Toshihide.

Research output: Article

*Combinatorica*, vol. 22, no. 4, pp. 537-554. https://doi.org/10.1007/s00493-002-0006-8

}

TY - JOUR

T1 - Convex embeddings and bisections of 3-connected graphs

AU - Nagamochi, Hiroshi

AU - Jordán, T.

AU - Nakao, Yoshitaka

AU - Ibaraki, Toshihide

PY - 2002

Y1 - 2002

N2 - Given two disjoint subsets T1 and T2 of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where |T1| and |T2| are even numbers, we show that V can be partitioned into two sets V1 and V2 such that the graphs induced by V1 and V2 are both connected and |V1∩Tj| = |V2∩Tj| = |Tj|/2 holds for each j = 1, 2. Such a partition can be found in O(|V|2log|V|) time. Our proof relies on geometric arguments. We define a new type of 'convex embedding' of k-connected graphs into real space Rk-1 and prove that for k = 3 such an embedding always exists.

AB - Given two disjoint subsets T1 and T2 of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where |T1| and |T2| are even numbers, we show that V can be partitioned into two sets V1 and V2 such that the graphs induced by V1 and V2 are both connected and |V1∩Tj| = |V2∩Tj| = |Tj|/2 holds for each j = 1, 2. Such a partition can be found in O(|V|2log|V|) time. Our proof relies on geometric arguments. We define a new type of 'convex embedding' of k-connected graphs into real space Rk-1 and prove that for k = 3 such an embedding always exists.

UR - http://www.scopus.com/inward/record.url?scp=0037000918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037000918&partnerID=8YFLogxK

U2 - 10.1007/s00493-002-0006-8

DO - 10.1007/s00493-002-0006-8

M3 - Article

AN - SCOPUS:0037000918

VL - 22

SP - 537

EP - 554

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -