Almost all string graphs are intersection graphs of plane convex sets

TL;DR

This paper proves that almost all string graphs can be represented as intersection graphs of plane convex sets, revealing a structural property and confirming a conjecture about their geometric nature.

Contribution

It establishes that nearly all string graphs have a five-clique partition with a non-adjacent pair and shows these graphs are intersection graphs of plane convex sets.

Findings

Almost all string graphs can be partitioned into five cliques with a non-connected pair.

Every graph with such a partition is an intersection graph of plane convex sets.

Almost all string graphs are intersection graphs of plane convex sets.

Abstract

A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of {\em almost all} string graphs on $n$ vertices can be partitioned into {\em five} cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that {\em almost all} string graphs on $n$ vertices are intersection graphs of plane convex sets.