The $2nd-$convex hull of every optimal rectilinear drawing of $K_{n}$ is a triangle

TL;DR

This paper proves that for complete graphs with at least 8 vertices, the second convex hull in any optimal rectilinear drawing always forms a triangle, revealing a consistent geometric property of such drawings.

Contribution

It establishes a universal geometric property of optimal rectilinear drawings of complete graphs for all n ≥ 8, specifically regarding the shape of the second convex hull.

Findings

The second convex hull is always a triangle for n ≥ 8.

This property holds for all optimal rectilinear drawings of K_n.

The result deepens understanding of geometric structure in graph drawings.

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 $n\geq 8$, the second convex hull of every optimal rectilinear drawing of the complete graph $K_n$ is a triangle.