Upper Bounds on the Number of Vertices of Weight <=k in Particular Arrangements of Pseudocircles
Ronald Ortner · arXiv (Cornell University) · 2007
Abstract. In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight ≤ k in certain arrangements of pseudocircles in the plane. In particular, forbidding certain subarrangements we improve the known bound of 6n − 12 (cf. [2]) for vertices of weight 0 in arrangements of n pseudocircles to 4n − 6. In complete arrangements (i.e. arrangements with each two pseudocircles intersecting) we identify two subarrangements of three and four pseudocircles, respectively, whose absence gives improved bounds for vertices of weight 0 and more generally for vertices of weight ≤ k. 1.