Abstract
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(m klog m) on a graph with m edges.
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M.J. Pelsmajer partially supported by NSA Grant H98230-08-1-0043.
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Pelsmajer, M.J., Schaefer, M. & Štefankovič, D. Crossing Numbers of Graphs with Rotation Systems. Algorithmica 60, 679–702 (2011). https://doi.org/10.1007/s00453-009-9343-y
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DOI: https://doi.org/10.1007/s00453-009-9343-y