Journal of Graph Algorithms and Applications
http://jgaa.info ISSN: 15261719
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DOI: 10.7155/jgaa.00266 Adjacent Crossings Do Matter Radoslav Fulek , Michael J. Pelsmajer , Marcus Schaefer , and Daniel Štefankovič To be published in the Special Issue of GD 2011. Abstract In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if the two edges share no endpoint. For a graph G, let ocr(G) be the smallest number odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G)<ocr(G), answering an open question of Székely. The same graph G also separates two notions of algebraic crossing numbers that Tutte expected to be the same. The graph G was found via considering monotone drawings of ordered graphs. A drawing of a graph is xmonotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. In an ordered graph, the vertices have a lefttoright ordering that must be preserved in xmonotone drawings. For every integer n>0 we construct an ordered graph G such that for xmonotone drawings, the monotone variant of ocr and iocr satisfy 2=moniocr(G)<= monocr(G)n$. We can also separate monocr from its variant in which crossings of adjacent edges are prohibited. We also offer a general translation result from monotone separations to nonmonotone separations. This could prove useful in settling several open separation problems, such as pair crossing number versus crossing number.

Posted here: May 2012. 
Journal of Graph Algorithms and Applications 