Simultaneous Geometric Graph Embeddings
Estrella-Balderrama, Alejandro and Gassner, Elisabeth and Jünger, Michael and Percan, Merijam and Schaefer, Marcus and Schulz, Michael (2008) Simultaneous Geometric Graph Embeddings. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 280-290 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_28).
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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  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 long-standing open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE.
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