, Volume 60, Issue 3, pp 679–702

Crossing Numbers of Graphs with Rotation Systems

  • Michael J. Pelsmajer
  • Marcus Schaefer
  • Daniel Štefankovič

DOI: 10.1007/s00453-009-9343-y

Cite this article as:
Pelsmajer, M.J., Schaefer, M. & Štefankovič, D. Algorithmica (2011) 60: 679. doi:10.1007/s00453-009-9343-y


We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněný’s result that computing the crossing number of a cubic graph is NP-complete.

We also consider the special case of multigraphs with rotation systems on a fixed number k of vertices. For k=1 we give an O(mlog m) algorithm, where m is the number of edges, and for loopless multigraphs on 2 vertices we present a linear time 2-approximation algorithm. In both cases there are interesting connections to edit-distance problems on (cyclic) strings. For larger k we show how to approximate the crossing number to within a factor of \({k+4\choose4}/5\) in time O(mklog m) on a graph with m edges.


Crossing number Rotation system Odd crossing number Independent odd crossing number Tournaments NP-completeness 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Michael J. Pelsmajer
    • 1
  • Marcus Schaefer
    • 2
  • Daniel Štefankovič
    • 3
  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of Computer ScienceDePaul UniversityChicagoUSA
  3. 3.Computer Science DepartmentUniversity of RochesterRochesterUSA

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