Crossing Number of Graphs with Rotation Systems

  • Michael J. Pelsmajer
  • Marcus Schaefer
  • Daniel Štefankovič
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

We show that computing the crossing number of a graph with a given rotation system is NP-complete. This result leads to a new and much simpler proof of Hliněný’s result, that computing the crossing number of a cubic graph (without rotation system) is NP-complete. We also investigate the special case of multigraphs with rotation systems on a fixed number k of vertices. For k = 1 and k = 2 the crossing number can be computed in polynomial time and approximated to within a factor of 2 in linear time. For larger k we show how to approximate the crossing number to within a factor of \({k+4\choose 4}/5\) in time O(m k + 2) on a graph with m edges.

Keywords

crossing number computational complexity computational geometry 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael J. Pelsmajer
    • 1
  • Marcus Schaefer
    • 2
  • Daniel Štefankovič
    • 3
  1. 1.Illinois Institute of TechnologyChicagoUSA
  2. 2.DePaul UniversityChicagoUSA
  3. 3.University of RochesterRochesterUSA

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