## Abstract

A plane graph is a graph embedded in a plane without edge crossings. Fáry's theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fáry's theorem to a class of non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. Further, we show that there are 1-plane graphs for which every straight-line drawing has exponential area. To the best of our knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.

Original language | English |
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Pages (from-to) | 335-346 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science |

Volume | 7434 LNCS |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |

Event | 18th Annual International Computing and Combinatorics Conference, COCOON 2012 - Sydney, NSW, Australia Duration: 20 Aug 2012 → 22 Aug 2012 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)