String Graphs: Product Structure and Localised Representations

Abstract

We investigate string graphs through the lens of graph product structure theory, which describes complicated graphs as subgraphs of strong products of simpler building blocks. A graph $G$ is called a string graph if its vertices can be represented by a collection $\mathcal{C}$ of continuous curves (called a string representation of $G$) in a surface so that two vertices are adjacent in $G$ if and only if the corresponding curves in $\mathcal{C}$ cross. We prove that every string graph with bounded maximum degree in a fixed surface is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This extends recent product structure theorems for string graphs. Applications of this result are presented. This product structure theorem ceases to be true if the `bounded maximum degree' assumption is relaxed to `bounded degeneracy'. For string graphs in the plane, we give an alternative proof of this result. Specifically, we show that every string graph in the plane has a `localised' string representation where the number of crossing points on the curve representing a vertex $u$ is bounded by a function of the degree of $u$. Our proof of the product structure theorem also leads to a result about the treewidth of outerstring graphs, which qualitatively extends a result of Fox and Pach [Eur. J. Comb. 2012] about outerstring graphs with bounded maximum degree. We extend our result to outerstring graphs defined in arbitrary surfaces.