Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard

@article{Cabello2013AddingOE,
  title={Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard},
  author={Sergio Cabello and Bojan Mohar},
  journal={SIAM J. Comput.},
  year={2013},
  volume={42},
  pages={1803-1829}
}
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its… 
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