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 they share no endpoint. For a graph G, let ocr(G) be the smallest number of 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 a question by Szekely, and--for the first time--giving evidence that crossings of adjacent edges may not always be trivial to eliminate. The graph G is based on a separation of iocr and ocr for monotone drawings of ordered graphs. A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. A graph is ordered if each of its vertices is assigned a distinct x-coordinate. We construct a family of ordered graphs such that for x-monotone drawings, the monotone variants of ocr and iocr satisfy mon-iocr(G)<O(mon−iocr(G)1/2).
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