On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
On levels in arrangements of surfaces in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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Pinchasi and Radoičić [1] 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 C4 has O(n3/2log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.