# String Graph Obstacles of High Girth and of Bounded Degree

## TL;DR

This paper investigates obstacles to string graphs with restrictions on girth and degree, constructing new examples and characterizing subcubic obstacles, leading to efficient recognition algorithms for certain classes.

## Contribution

It introduces new obstacle constructions with restricted girth and degree, and characterizes subcubic string graphs, enabling linear-time recognition algorithms.

## Key findings

- Constructed an infinite family of obstacles of girth four
- Proved existence of a subcubic obstacle of girth three
- Developed a linear-time recognition algorithm for subcubic string graphs of bounded treewidth

## Abstract

A string graph is the intersection graph of curves in the plane. Kratochv\'il previously showed the existence of infinitely many obstacles: graphs that are not string graphs but for which any edge contraction or vertex deletion produces a string graph. Kratochv\'il's obstacles contain arbitrarily large cliques, so they have girth three and unbounded degree. We extend this line of working by studying obstacles among graphs of restricted girth and/or degree. We construct an infinite family of obstacles of girth four; in addition, our construction is $K_{2,3}$-subgraph-free and near-planar (planar plus one edge). Furthermore, we prove that there is a subcubic obstacle of girth three, and that there are no subcubic obstacles of high girth. We characterize the subcubic string graphs as having a matching whose contraction yields a planar graph, and based on this characterization we find a linear-time algorithm for recognizing subcubic string graphs of bounded treewidth.