A Van Kampen type obstruction for string graphs

TL;DR

This paper introduces a new algebraic obstruction, inspired by Van Kampen's, to determine if a graph is a string graph, providing a characterization that is necessary but not sufficient.

Contribution

It establishes a Van Kampen type algebraic obstruction for string graphs and explores its limitations compared to planarity.

Findings

A graph is a string graph iff it admits a certain planar drawing.

The algebraic obstruction must vanish for string graphs.

The obstruction is necessary but not sufficient for string graph recognition.

Abstract

In this paper we prove that a graph is a string graph (the intersection graph of curves in the plane) if and only if it admits a drawing in the plane with certain properties. This also allows us to define an algebraic obstruction, similar to the Van Kampen obstruction to embeddability, which must vanish for every string graph. However, unlike in the case of graph planarity this obstruction is incomplete.