Adjacent Crossings Do Matter
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 Székely, 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).
- 2.Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings, and level-planarity. Accepted for WG (2011)Google Scholar
- 19.Valtr, P.: On the pair-crossing number. In: Combinatorial and Computational Geometry. Math. Sci. Res. Inst. Publ., vol. 52, pp. 569–575. Cambridge University Press, Cambridge (2005)Google Scholar
- 20.West, D.: Open problems - graph theory and combinatorics, http://www.math.uiuc.edu/~west/openp/ (accessed April 7, 2005)