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- W4287017524 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 $e^{Oleft(sqrt{k}right)}$ 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 $Obig(sqrt{k}big)$ is asymptotically tight. The previous upper bound bound $2^{(2k)^4}$ was proved by Pach, Tardos, and T'oth [Graph Drawing 2020]. We prove the above result by proving the asymptotic upper bound $e^{Oleft(sqrt{k}right)}$ for a similar problem when $x in V_n$, and by proving a close relation between the two problems." @default.
- W4287017524 created "2022-07-25" @default.
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- W4287017524 date "2021-08-31" @default.
- W4287017524 modified "2023-09-26" @default.
- W4287017524 title "Non-homotopic Loops with a Bounded Number of Pairwise Intersections" @default.
- W4287017524 doi "https://doi.org/10.48550/arxiv.2108.13953" @default.
- W4287017524 hasPublicationYear "2021" @default.
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