Faces in rectilinear drawings of complete graphs

TL;DR

This paper explores extremal face configurations in convex rectilinear drawings of complete graphs, establishing conditions for convex polygons as faces and characterizing regular drawings with convex 5-gon faces.

Contribution

It introduces the first study of face extremal problems in convex rectilinear drawings of complete graphs and characterizes when regular drawings contain convex 5-gon faces.

Findings

If no common interior point of three edges exists, a convex 5-gon face always exists.

There are convex rectilinear drawings without any convex k-gon face for k ≥ 6.

Characterization of regular drawings containing convex 5-gon faces.

Abstract

We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of $K_n$ does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex $k$-gon with $k \geq 6$. A convex rectilinear drawing of $K_n$ is \emph{regular} if its vertices correspond to vertices of a regular convex $n$-gon. We characterize positive integers $n$ for which regular drawings of $K_n$ contain a face forming a convex 5-gon. To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.