… The core result of this paper, Theorem 2.1, shows that we can remove even crossings
without introducing any odd crossings. With this tool at hand, we were able to give new and …

… remove all crossings with even edges. From this we can conclude that cr S ( G ) , the crossing
… ) 2 , where ocr S ( G ) is the odd crossing number of G on surface S . Finally, we prove that …

… as to remove all crossings with even edges (an edge is even if it crosses every other edge
an even number of times). From this result we can conclude that crS(G), the crossing number …

… even and we can apply Lemma 2.2 to remove all crossings with edges of C without
changing the rotation system or adding new pairs of edges that cross oddly. Now C is …

… This number is a lower bound to the optimum and may even grow exponentially … crossings
is very useful even on dense instances. We also observed that drawings with more crossings …

… Consider an even loop e at u. If an odd loop f crosses e, we simply remove the segment
between the first and last crossing with e and directly connect the two ends alongside the …

… But k can be as large as n 2− 1 for even n, which is the case … non-crossing spanning tree
after removing any self-crossing 2-… crossing subtree that spans n − k vertices after removing ⌈kn …

… And in spite of the fact that determining the independent odd crossing number of a graph
is NP-complete [9], we feel that due to its algebraic nature it offers an intriguing and …

… of the inner framework cross an even number of times. We next show how to remove crossings
between edges of the inner framework. To this end, let us consider Q1 = u1v1w1. Let e be …

… We therefore expect that, at large scales, zero crossings may be created even for
Gaussian filters, unless the image is very simple (for instance an isolated, straight step edge). …