### Abstract

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K _{5} or K _{3,3}. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE.

Original language | English (US) |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 34th International Workshop, WG 2008, Revised Papers |

Pages | 146-158 |

Number of pages | 13 |

DOIs | |

State | Published - Dec 1 2008 |

Event | 34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008 - Durham, United Kingdom Duration: Jun 30 2008 → Jul 2 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5344 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008 |
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Country | United Kingdom |

City | Durham |

Period | 6/30/08 → 7/2/08 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph-Theoretic Concepts in Computer Science - 34th International Workshop, WG 2008, Revised Papers*(pp. 146-158). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5344 LNCS). https://doi.org/10.1007/978-3-540-92248-3_14