GD 2007: Graph Drawing pp 3-12

# Crossing Number of Graphs with Rotation Systems

• Michael J. Pelsmajer
• Marcus Schaefer
• Daniel Štefankovič
Conference paper
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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## 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