Crossing Numbers and Parameterized Complexity
Pelsmajer, Michael J. and Schaefer, Marcus and Stefankovic, Daniel (2008) Crossing Numbers and Parameterized Complexity. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 31-36 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_6).
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The odd crossing number of G is the smallest number of pairs of edges that cross an odd number of times in any drawing of G. We show that there always is a drawing realizing the odd crossing number of G that uses at most 9^k crossings, where k is the odd crossing number of G. As a consequence of this and a result of Grohe we can show that the odd crossing number is fixed-parameter tractable.
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