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Non-homotopic Loops with a Bounded Number of Pairwise Intersections
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SYSNO ASEP 0551784 Document Type C - Proceedings Paper (int. conf.) R&D Document Type Conference Paper Title Non-homotopic Loops with a Bounded Number of Pairwise Intersections Author(s) Blažej, V. (CZ)
Opler, M. (CZ)
Šileikis, Matas (UIVT-O) RID, ORCID, SAI
Valtr, P. (CZ)Number of authors 4 Source Title Graph Drawing and Network Visualization. 29th International Symposium GD 2021, Revised Selected Papers. - Cham : Springer, 2021 / Purchase H. C. ; Rutter I. - ISSN 0302-9743 - ISBN 978-3-030-92930-5 Pages s. 210-222 Number of pages 13 s. Publication form Print - P Action GD 2021: International Symposium on Graph Drawing and Network Visualization /29./ Event date 14.09.2021 - 17.09.2021 VEvent location Tübingen Country DE - Germany Event type WRD Language eng - English Country CH - Switzerland Keywords Graph drawing ; Non-homotopic loops ; Curve intersections ; Plane OECD category Pure mathematics R&D Projects GJ20-27757Y GA ČR - Czech Science Foundation (CSF) Institutional support UIVT-O - RVO:67985807 EID SCOPUS 85122149272 DOI 10.1007/978-3-030-92931-2_15 Annotation Let V_n be a set of n points in the plane and let x∈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 Vn . For n=2, we give an upper bound e^O(k^(1/2)) 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(k^(1/2)) is asymptotically tight. The previous upper bound 2^((2k)^4) was proved by Pach et al. [6]. We prove the above result by proving the asymptotic upper bound e^O(k^(1/2)) for a similar problem when x∈V_n, and by proving a close relation between the two problems. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2022
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