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.
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Pelsmajer, M.J., Schaefer, M., Štefankovič, D. (2008). Crossing Number of Graphs with Rotation Systems. In: Hong, SH., Nishizeki, T., Quan, W. (eds) Graph Drawing. GD 2007. Lecture Notes in Computer Science, vol 4875. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77537-9_3
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DOI: https://doi.org/10.1007/978-3-540-77537-9_3
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