Holes in Convex and Simple Drawings

TL;DR

This paper explores the concept of holes in simple graph drawings, introducing new notions, generalizing existing theorems, and analyzing structural properties of convex and pseudolinear drawings.

Contribution

It introduces the notion of $k$-holes in simple drawings, generalizes known theorems, and provides structural insights and constructions related to empty cycles.

Findings

Existence of empty 4-cycles in every simple drawing of $K_n$

Construction with only $ heta(n^2)$ empty 4-cycles

A family of simple drawings without 4-holes

Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. We present a family of simple drawings without $4$-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step is the structural investigation of pseudolinear subdrawings in convex drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $\Theta(n^2)$ of them.