Geodesic
Empty Triangles in Generalized Twisted Drawings of $K_n$
Journal of Graph Algorithms and Applications, Special issue on Selected papers from the Thirtieth International Symposium on Graph Drawing and Network Visualization, GD 2022 , Tome 27 (2023) no. 8, pp. 721-735.

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Simple drawings are drawings of graphs in the plane such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once, and there is a ray emanating from $O$ which crosses every edge exactly once. We show that all generalized twisted drawings of $K_n$ contain exactly $2n-4$ empty triangles, by this making a substantial step towards proving the conjecture that any simple drawing of $K_n$ contains at least $2n-4$ empty triangles. 
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     author = {Alfredo Garc{\'\i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
     title = {Empty {Triangles} in {Generalized} {Twisted} {Drawings} of $K_n$},
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Alfredo García; Javier Tejel; Birgit Vogtenhuber; Alexandra Weinberger. Empty Triangles in Generalized Twisted Drawings of $K_n$. Journal of Graph Algorithms and Applications, 
							Special issue on Selected papers from the Thirtieth International Symposium on Graph Drawing and Network Visualization, GD 2022
					, Tome 27 (2023) no. 8, pp. 721-735. doi : 10.7155/jgaa.00637. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00637/

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