### Abstract

We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.

Original language | English |
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Pages (from-to) | 427-452 |

Number of pages | 26 |

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 Mar 1 |

### Keywords

- Biplane graphs
- Geometric graphs
- Graph augmentation
- k-connected graphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Graphs and Combinatorics*,

*31*(2), 427-452. https://doi.org/10.1007/s00373-015-1547-0