### 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) |
---|---|

Pages (from-to) | 385-398 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 44 |

Issue number | 8 |

DOIs | |

State | Published - Oct 2011 |

### Fingerprint

### Keywords

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

### 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*,

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

**Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges.** / Fowler, J. Joseph; Jünger, Michael; Kobourov, Stephen G; Schulz, Michael.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 44, no. 8, pp. 385-398. https://doi.org/10.1016/j.comgeo.2011.02.002

}

TY - JOUR

T1 - Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

AU - Fowler, J. Joseph

AU - Jünger, Michael

AU - Kobourov, Stephen G

AU - Schulz, Michael

PY - 2011/10

Y1 - 2011/10

N2 - 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.

AB - 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.

KW - Graph drawing

KW - SEFE

KW - Simultaneous embedding

KW - Simultaneous embedding with fixed edges

UR - http://www.scopus.com/inward/record.url?scp=79953733331&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953733331&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2011.02.002

DO - 10.1016/j.comgeo.2011.02.002

M3 - Article

AN - SCOPUS:79953733331

VL - 44

SP - 385

EP - 398

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 8

ER -