### Abstract

A set of planar graphs {^{G1}(V,^{E1}),⋯, ^{Gk}(V,^{Ek})} admits a simultaneous embedding if they can be drawn on the same pointset P of order n in the Euclidean plane such that each point in P corresponds one-to-one to a vertex in V and each edge in ^{Ei} does not cross any other edge in ^{Ei} (except at endpoints) for i∈{1,⋯,k}. A fixed edge is an edge (u,v) that is drawn using the same simple curve for each graph ^{Gi} whose edge set ^{Ei} contains the edge (u,v). We give a necessary and sufficient condition for two graphs whose union is homeomorphic to ^{K5} or K3 _{,3} to admit a simultaneous embedding with fixed edges (SEFE). 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 O(^{n4})-time algorithms to compute a SEFE.

Original language | English (US) |
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Pages (from-to) | 385-398 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 44 |

Issue number | 8 |

DOIs | |

State | Published - Oct 1 2011 |

### Keywords

- Graph drawing
- SEFE
- Simultaneous embedding
- Simultaneous embedding with fixed edges

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Computational Geometry: Theory and Applications*,

*44*(8), 385-398. https://doi.org/10.1016/j.comgeo.2011.02.002