Upper Bounds on the Number of Vertices of Weight <=k in Particular Arrangements of Pseudocircles

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. Kedem et al., 1986) 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.