Geodesic
On the number of intersections of two polygons
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 217-228.

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We study the maximum possible number $f(k,l)$ of intersections of the boundaries of a simple $k$-gon with a simple $l$-gon in the plane for $k,l\ge 3$. To determine the number $f(k,l)$ is quite easy and known when $k$ or $l$ is even but still remains open for $k$ and $l$ both odd. We improve (for $k\le l$) the easy upper bound $kl-l$ to $kl-\left\lceil k/6\right\rceil-l$ and obtain exact bounds for $k=5$ $(f(5,l)=4l-2)$ in this case.
Classification : 52C10, 52C45
Keywords: geometry; polygon; intersection; combinatorial complexity 
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     title = {On the number of intersections of two polygons},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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Černý, Jakub; Kára, Jan; Král', Daniel; Podbrdský, Pavel; Sotáková, Miroslava; Šámal, Robert. On the number of intersections of two polygons. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 217-228. http://geodesic.mathdoc.fr/item/CMUC_2003__44_2_a2/