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 long-standing open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. © 2008 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Estrella-Balderrama, A., Gassner, E., Jünger, M., Percan, M., Schaefer, M., & Schulz, M. (2008). Simultaneous geometric graph embeddings. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4875 LNCS, pp. 280–290). https://doi.org/10.1007/978-3-540-77537-9_28
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