Non-homotopic Loops with a Bounded Number of Pairwise Intersections

Abstract

Let $V_n$ be a set of $n$ points in the plane and let $x\notin V_n$. An $x$-loop is a continuous closed curve not containing any point of $V_n$. We say that two $x$-loops are non-homotopic if they cannot be transformed continuously into each other without passing through a point of $V_n$. For $n=2$, we give an upper bound $e^{O\left(\sqrt{k}\right)}$ on the maximum size of a family of pairwise non-homotopic $x$-loops such that every loop has fewer than $k$ self-intersections and any two loops have fewer than $k$ intersections. The exponent $O\left(\sqrt{k}\right)$ is asymptotically tight. The previous upper bound bound $2^{(2k{)}^4}$ was proved by Pach, Tardos, and Tóth [Graph Drawing 2020]. We prove the above result by proving the asymptotic upper bound $e^{O\left(\sqrt{k}\right)}$ for a similar problem when $x\in V_n$, and by proving a close relation between the two problems.

Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)