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

TL;DR

This paper establishes improved upper bounds on the number of vertices with weight less than or equal to k in specific arrangements of pseudocircles, especially when certain subarrangements are forbidden, refining previous bounds.

Contribution

It introduces new upper bounds for vertices of weight <=k in pseudocircle arrangements by excluding specific subarrangements, advancing understanding of their combinatorial limits.

Findings

Improved bound of 4n-6 for vertices of weight 0 when certain subarrangements are forbidden.

Identification of specific subarrangements whose absence leads to tighter bounds.

Extension of bounds to vertices of weight <=k in complete arrangements.

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.