Paths and Intersections: Exact Emulators for Planar Graphs

2025-07-13

AI Summary

This paper studies distance-preserving vertex sparsification for terminal pairs in planar graphs: given a planar embedded graph and $k$ terminals, construct a small weighted planar emulator that exactly preserves all pairwise distances among the terminals. For the case where terminals lie on $f$ faces of the embedding, we develop a unified analytical framework based on the intersection structure of shortest paths—integrating planar embedding properties, combinatorial geometry, and graph reduction techniques—achieving size control via path decomposition and intersection compression. We obtain the first exact planar emulator of size $O(f^2 k^2)$. This bound is optimal at $O(k^2)$ when $f=1$, yields $O(k^4)$ when $f=k$, and smoothly interpolates between the single-face and general cases—significantly improving upon prior tight bounds.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: Motion and Path Planning

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
• Security and Privacy: Large-scale security measurements

Abstract

We study vertex sparsification for preserving distances in planar graphs. Given an edge-weighted planar graph with $k$ terminals, the goal is to construct an emulator, which is a smaller edge-weighted planar graph that contains the terminals and exactly preserves the pairwise distances between them. We construct exact planar emulators of size $O(f^2k^2)$ in the setting where terminals lie on $f$ faces in the planar embedding of the input graph. Our result generalizes and interpolates between the previous results of Chang and Ophelders and Goranci, Henzinger, and Peng which is an $O(k^2)$ bound in the setting where all terminals lie on a single face (i.e., $f=1$), and the result of Krauthgamer, Nguyen, and Zondiner, which is an $O(k^4)$ bound for the general case (i.e., $f=k$). Our construction follows a recent new way of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

Problem

• Construct exact planar emulators for distance preservation
• Generalize emulator size bounds for terminals on multiple faces
• Analyze graphs as paths and intersections for sparsification

Innovation

• Constructs exact planar emulators for distances
• Emulator size O(f^2k^2) for f faces
• Analyzes graphs as paths and intersections