On a cyclic string-to-string correction problem
Information Processing Letters
An Extension of the String-to-String Correction Problem
Journal of the ACM (JACM)
The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information
The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information
On the complexity of the Extended String-to-String Correction Problem
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Crossing number is hard for cubic graphs
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B
Odd crossing number is not crossing number
GD'05 Proceedings of the 13th international conference on Graph Drawing
Improved upper bounds on the crossing number
Proceedings of the twenty-fourth annual symposium on Computational geometry
Unexpected behaviour of crossing sequences
Journal of Combinatorial Theory Series B
Removing independently even crossings
GD'09 Proceedings of the 17th international conference on Graph Drawing
Complexity of some geometric and topological problems
GD'09 Proceedings of the 17th international conference on Graph Drawing
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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 Hlineny'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 4)/5 in time O(mk+2) on a graph with m edges.