String Graph Obstacles of High Girth and of Bounded Degree

2025-08-29

AI Summary

This study investigates minimal obstructions to string graphs—non-string graphs that become string graphs upon any edge contraction or vertex deletion. Under constraints on girth and maximum degree, we systematically characterize their existence and structure. First, we construct an infinite family of minimal obstructions with girth 4, excluding $K_{2,3}$ and being nearly planar. Second, we prove that subcubic minimal obstructions of girth 3 exist, whereas none exist for girth $geq 5$. Third, we establish a structural characterization of subcubic string graphs via “planar-contractible matchings.” Leveraging this characterization, we design a linear-time algorithm to recognize subcubic string graphs on graphs of bounded treewidth. Finally, we fully classify the existence and parameter thresholds of all minimal obstructions in the subcubic case, determining precisely which girth and forbidden subgraph constraints admit or preclude such obstructions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A string graph is the intersection graph of curves in the plane. Kratochvíl 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. Kratochvíl'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.

Problem

• Constructing infinite family of high girth string graph obstacles
• Proving existence of subcubic girth three string obstacles
• Characterizing subcubic string graphs through matching contraction

Innovation

• Constructed infinite girth-four obstacle family
• Proved subcubic girth-three obstacle existence
• Developed linear-time subcubic recognition algorithm