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Special issue on Selected papers from the Twenty-eighth International Symposium on Graph Drawing and Network Visualization, GD 2020
DOI: 10.7155/jgaa.00600
On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens
Vol. 26, no. 3, pp. 381-399, 2022. Regular paper.
Abstract A star-simple drawing of a graph is a drawing in which adjacent edges
do not cross. In contrast, there is no restriction on the number of
crossings between two independent edges. We forbid empty lenses, i.e.,
every lens is required to enclose a vertex, and show that with this
restriction $3\cdot(n-4)!$ is an upper bound on the number of crossings between two edges of a star-simple
drawing of $K_n$. It follows that $n!$ bounds the total number
of crossings in the drawing. This is the first finite upper
bound on the number of crossings in star-simple drawings of the complete
graph $K_n$ with no empty lens. For a lower bound we construct a
star-simple drawing of $K_n$ with no empty lens in which a pair of edges
contributes $5^{n/2-2}$ crossings.
This work is licensed under the terms of the CC-BY license.
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Submitted: May 2021.
Reviewed: April 2022.
Revised: May 2022.
Accepted: June 2022.
Final: June 2022.
Published: June 2022.
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