### Abstract

A geometric simultaneous embedding of two graphs G _{1}∈= ∈(V _{1},E _{1}) and G _{2}∈=∈(V _{2},E _{2}) with a bijective mapping of their vertex sets γ: V _{1} →V _{2} is a pair of planar straight-line drawings Γ _{1} of G _{1} and Γ _{2} of G _{2}, such that each vertex v _{2}∈=∈γ(v _{1}) is mapped in Γ _{2} to the same point where v _{1} is mapped in Γ _{1}, where v _{1}∈ ∈V _{1} and v _{2}∈ ∈V _{2}. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not mapped exactly to the same but to "near" points in the different drawings.

Original language | English (US) |
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Title of host publication | Graph Drawing - 15th International Symposium, GD 2007, Revised Papers |

Pages | 268-279 |

Number of pages | 12 |

DOIs | |

State | Published - Aug 27 2008 |

Event | 15th International Symposium on Graph Drawing, GD 2007 - Sydney, Australia Duration: Sep 24 2007 → Sep 26 2007 |

### 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 | 4875 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th International Symposium on Graph Drawing, GD 2007 |
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Country | Australia |

City | Sydney |

Period | 9/24/07 → 9/26/07 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph Drawing - 15th International Symposium, GD 2007, Revised Papers*(pp. 268-279). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4875 LNCS). https://doi.org/10.1007/978-3-540-77537-9_27