Non-homotopic Loops with a Bounded Number of Pairwise Intersections

Autoři: Matas Šileikis, Ph.D.
Rok vydání: 2021
Typ publikace: Konferenční příspěvek (mezinárodní konf.)
Periodikum: Graph Drawing and Network Visualization. 29th International Symposium GD 2021, Revised Selected Papers
Projekty: Náhodné diskrétní struktury
Bibliografická citace: BLAŽEJ, V., OPLER, M., ŠILEIKIS, Matas, VALTR, P. Non-homotopic Loops with a Bounded Number of Pairwise Intersections. In: PURCHASE, H. C., RUTTER, I., eds. Graph Drawing and Network Visualization. 29th International Symposium GD 2021, Revised Selected Papers. Cham: Springer, 2021, S. 210-222. Lecture Notes in Computer Science, 12868. ISBN 978-3-030-92930-5. ISSN 0302-9743
Dostupné z: doi:10.1007/978-3-030-92931-2_15
DOI: 10.1007/978-3-030-92931-2_15
ID: 551784

Abstrakt

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.