Computing crossing numbers in quadratic time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Crossing number is hard for cubic graphs
Journal of Combinatorial Theory Series B
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the crossing number of almost planar graphs
GD'06 Proceedings of the 14th international conference on Graph drawing
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Crossing Number and Weighted Crossing Number of Near-Planar Graphs
Algorithmica - Special issue: Algorithms, Combinatorics, & Geometry
Planar crossing numbers of genus g graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Do we really understand the crossing numbers?
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
An algorithm for the graph crossing number problem
Proceedings of the forty-third annual ACM symposium on Theory of computing
A tighter insertion-based approximation of the crossing number
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Vertex insertion approximates the crossing number of apex graphs
European Journal of Combinatorics
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show that it is NP-hard to compute the crossing number of near-planar graphs. The main idea in the reduction is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This approach can be used to prove hardness of some other geometric problems. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hlinený.