Intersection Reverse Sequences and Geometric Applications

13 years 3 months ago
Intersection Reverse Sequences and Geometric Applications
Pinchasi and Radoiˇci´c [11] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved 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-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.
Adam Marcus, Gábor Tardos
Added 01 Jul 2010
Updated 01 Jul 2010
Type Conference
Year 2004
Where GD
Authors Adam Marcus, Gábor Tardos
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