Simultaneous geometric graph embeddings

Authors:
Alejandro Estrella-Balderrama;Elisabeth Gassner;Michael Jünger;Merijam Percan;Marcus Schaefer;Michael Schulz
Affiliations:
Department of Computer Science, University of Arizona;Institut für Mathematik B, Technische Universität Graz;Institut für Informatik, Universität zu Köln;Institut für Informatik, Universität zu Köln;School of CTI, DePaul University;Institut für Informatik, Universität zu Köln
Venue:
GD'07 Proceedings of the 15th international conference on Graph drawing
Year:
2007

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Abstract

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.