### 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 v2=γ(v1), with v1 ∈ V _{1} and v2 ∈ V _{2}, is mapped in Γ _{2} to the same point where v1 is mapped in Γ _{1}. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that assuming that the input graphs do not share common edges does not yield larger 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 in the standard geometric simultaneous embedding setting. Finally, we present some results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly at the same, but just at 'nearby' points in different drawings.

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

Pages (from-to) | 447-465 |

Number of pages | 19 |

Journal | Journal of Graph Algorithms and Applications |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics
- Computer Science(all)
- Computer Science Applications

### Cite this

*Journal of Graph Algorithms and Applications*,

*13*(3), 447-465. https://doi.org/10.7155/jgaa.00194

**Constrained simultaneous and near-simultaneous embeddings.** / Frati, Fabrizio; Kaufmann, Michael; Kobourov, Stephen G.

Research output: Contribution to journal › Article

*Journal of Graph Algorithms and Applications*, vol. 13, no. 3, pp. 447-465. https://doi.org/10.7155/jgaa.00194

}

TY - JOUR

T1 - Constrained simultaneous and near-simultaneous embeddings

AU - Frati, Fabrizio

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

PY - 2009

Y1 - 2009

N2 - 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 v2=γ(v1), with v1 ∈ V 1 and v2 ∈ V 2, is mapped in Γ 2 to the same point where v1 is mapped in Γ 1. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that assuming that the input graphs do not share common edges does not yield larger 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 in the standard geometric simultaneous embedding setting. Finally, we present some results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly at the same, but just at 'nearby' points in different drawings.

AB - 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 v2=γ(v1), with v1 ∈ V 1 and v2 ∈ V 2, is mapped in Γ 2 to the same point where v1 is mapped in Γ 1. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that assuming that the input graphs do not share common edges does not yield larger 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 in the standard geometric simultaneous embedding setting. Finally, we present some results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly at the same, but just at 'nearby' points in different drawings.

UR - http://www.scopus.com/inward/record.url?scp=77951879330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951879330&partnerID=8YFLogxK

U2 - 10.7155/jgaa.00194

DO - 10.7155/jgaa.00194

M3 - Article

VL - 13

SP - 447

EP - 465

JO - Journal of Graph Algorithms and Applications

JF - Journal of Graph Algorithms and Applications

SN - 1526-1719

IS - 3

ER -