Intersection Reverse Sequences and Geometric Applications
Pinchasi and Radoičić  used the following observation to bound the number of edges in a topological graph with no self-intersecting cycles of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclicly according to the connecting edge, then the common elements in any two lists have reversed cyclic order. Building on their work we give an estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-intersecting C 4 has O(n 3/2log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n 3/2log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.
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