Simultaneous Geometric Graph Embeddings

  • Alejandro Estrella-Balderrama
  • Elisabeth Gassner
  • Michael Jünger
  • Merijam Percan
  • Marcus Schaefer
  • Michael Schulz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Alejandro Estrella-Balderrama
    • 1
  • Elisabeth Gassner
    • 3
  • Michael Jünger
    • 4
  • Merijam Percan
    • 4
  • Marcus Schaefer
    • 2
  • Michael Schulz
    • 4
  1. 1.Department of Computer ScienceUniversity of Arizona 
  2. 2.School of CTIDePaul University 
  3. 3.Institut für Mathematik BTechnische Universität Graz 
  4. 4.Institut für InformatikUniversität zu Köln 

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