O(1)-Distortion Planar Emulators for String Graphs

2025-10-24

AI Summary

This paper addresses the long-standing open problem of whether unweighted string graphs admit a planar emulator with constant distortion—i.e., whether one can construct a weighted planar graph on the same vertex set such that all pairwise shortest-path distances are preserved up to a constant multiplicative factor. We provide the first affirmative answer: every unweighted string graph admits an O(1)-distortion planar emulator. Our approach departs from classical embedding paradigms, instead leveraging structural properties of string graphs—specifically their intersection topology—and integrating tools from graph theory and combinatorial geometry to design a novel weighted planar construction. This result establishes the first strong distance equivalence between string graphs and planar graphs. It introduces a new paradigm for distance approximation in high-dimensional or non-planar graphs and has foundational implications for graph algorithms, network embedding, and metric compression.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

We show that every unweighted string graph $G$ has an $O(1)$-distortion planar emulator: that is, there exists an (edge-weighted) planar graph $H$ with $V(H) = V(G)$, such that every pair of vertices $(u,v)$ satisfies $delta_G(u,v) le delta_H(u,v) le O(1) cdot delta_G(u,v).$

Problem

• Constructing constant-distortion planar emulators for string graphs
• Preserving vertex distances between original and planar graphs
• Providing edge-weighted planar approximations for unweighted string graphs

Innovation

• Planar emulator with constant distortion
• Edge-weighted planar graph construction
• Distance preservation for string graphs