The $2nd-$convex hull of every optimal rectilinear drawing of $K_n$ is a triangle

Abstract

A rectilinear drawing of a graph $G$ is optimal if it has the smallest number of crossings among all rectilinear drawings of $G$. In this paper it is shown that for $n\ge 8$, the second convex hull of every optimal rectilinear drawing of the complete graph $K_n$ is a triangle.

arXiv commentThis paper has been withdrawn by the authors due to a crucial error in the proof of the main result