Almost All String Graphs are Intersection Graphs of Plane Convex Sets

Abstract

A 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 almost all string graphs on n vertices can be partitioned into 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 almost all string graphs on n vertices are intersection graphs of plane convex sets.