What induces plane structures in complete graph drawings?

TL;DR

This paper investigates the conditions under which certain plane structures emerge in complete graph drawings, proving unavoidable disjoint curves under specific rules and demonstrating constructions with controlled crossings.

Contribution

It provides new theoretical insights into the emergence of plane structures in complete graph drawings under various crossing rules.

Findings

Many disjoint curves are unavoidable under certain crossing constraints.

A construction exists where all curves have controlled crossing properties.

The paper characterizes plane structures guaranteed by different crossing rules.

Abstract

This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.