Forcing subarrangements in complete arrangements of pseudocircles

• Main Author: Ronald Ortner
• Format: Article
• Language: English
• Published: Carleton University 2015-09-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/37

In arrangements of <em>pseudocircles</em> (i.e., Jordan curves) the <em>weight</em> of a <em>vertex</em> (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in <em>complete</em> arrangements (in which each two pseudocircles intersect) $2n-1$ vertices of weight 0 force an <em>$\alpha$</em>-subarrangement, a certain arrangement of three pseudocircles. Similarly, $4n-5$ vertices of weight 0 force an $\alpha^4$-subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight $k$ for complete, $\alpha$-free and complete, $\alpha^4$-free arrangements. On the other hand, interpreting $\alpha$- and $\alpha^4$-arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.