### Abstract

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O( ^{n2})×O( ^{n2}) grid.

Original language | English (US) |
---|---|

Pages (from-to) | 117-130 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

### Fingerprint

### Keywords

- Graph drawing
- Planar graphs
- Simultaneous visualizations

### ASJC Scopus subject areas

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

### Cite this

*Computational Geometry: Theory and Applications*,

*36*(2), 117-130. https://doi.org/10.1016/j.comgeo.2006.05.006

**On simultaneous planar graph embeddings.** / Brass, Peter; Cenek, Eowyn; Duncan, Cristian A.; Efrat, Alon; Erten, Cesim; Ismailescu, Dan P.; Kobourov, Stephen G; Lubiw, Anna; Mitchell, Joseph S B.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 36, no. 2, pp. 117-130. https://doi.org/10.1016/j.comgeo.2006.05.006

}

TY - JOUR

T1 - On simultaneous planar graph embeddings

AU - Brass, Peter

AU - Cenek, Eowyn

AU - Duncan, Cristian A.

AU - Efrat, Alon

AU - Erten, Cesim

AU - Ismailescu, Dan P.

AU - Kobourov, Stephen G

AU - Lubiw, Anna

AU - Mitchell, Joseph S B

PY - 2007/2

Y1 - 2007/2

N2 - We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O( n2)×O( n2) grid.

AB - We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O( n2)×O( n2) grid.

KW - Graph drawing

KW - Planar graphs

KW - Simultaneous visualizations

UR - http://www.scopus.com/inward/record.url?scp=84867933017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867933017&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2006.05.006

DO - 10.1016/j.comgeo.2006.05.006

M3 - Article

AN - SCOPUS:84867933017

VL - 36

SP - 117

EP - 130

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 2

ER -