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.
Pelsmajer et al. showed a similar result for the odd crossing number REF .
Crossing numbers and parameterized complexity
Graph Drawing, volume 4875 of Lecture Notes in Computer Science
Mathematics Computer Science