Lecture Notes in Computer Science Volume 4875, 2008, pp 31-36

Crossing Numbers and Parameterized Complexity

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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.