Simultaneous Geometric Graph Embeddings

We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE.

• Type: Conference
• Year: 2007
• Where: GD
• Authors: Alejandro Estrella-Balderrama, Elisabeth Gassner, Michael Jünger, Merijam Percan, Marcus Schaefer, Michael Schulz
• Keywords: Computer Graphics, GD 2007, Graph Embedding, Rectilinear Crossing Number, Simultaneous Geometric Graph