An almost optimal bound on the number of intersections of two simple polygons

Ackerman, Eyal and Keszegh, Balázs and Rote, Günter (2020) An almost optimal bound on the number of intersections of two simple polygons. In: 36th International Symposium on Computational Geometry, SoCG 2020. Leibniz-Zentrum für Informatik, Wadern. ISBN 9783959771436

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Abstract

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dn6 e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C. © Eyal Ackerman, Balázs Keszegh, and Günter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).

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