Stabbing Faces By a Convex Curve

TL;DR

This paper proves that for any plane graph and any smooth convex curve not on a line, there exists a straight-line drawing where the curve crosses every face, revealing a new geometric property of plane graphs.

Contribution

It introduces a universal construction showing that any plane graph can be drawn so that a convex curve intersects all faces, a novel geometric insight.

Findings

Existence of such a drawing for all plane graphs

Construction method for the straight-line drawing

Implications for geometric graph theory

Abstract

We prove that, for every plane graph $G$ and every smooth convex curve $C$ not on a single line, there exists a straight-line drawing of $G$ for which every face is crossed by $C$.