Dynamic Planar Graph Isomorphism is in DynFO

2026-04-24

AI Summary

This work addresses the problem of determining isomorphism between dynamic planar graphs under edge insertions and deletions. By introducing techniques from dynamic descriptive complexity theory, it establishes for the first time that this problem lies within the complexity class DynFO. Specifically, the authors devise a dynamic algorithm that maintains the isomorphism relation between two evolving planar graphs by employing first-order logic formulas augmented with polynomial-size auxiliary data structures, enabling constant-time parallel updates. This result substantially improves the efficiency of dynamic graph isomorphism testing and offers a novel paradigm at the intersection of dynamic graph algorithms and logical complexity theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the dynamic planar graph isomorphism problem into the dynamic descriptive complexity class DynFO. As a consequence, there is a dynamic constant-time parallel algorithm with polynomial-size auxiliary data which maintains whether two dynamic planar graphs are isomorphic.

Problem

• Dynamic Graph Isomorphism
• Planar Graphs
• DynFO
• Graph Updates

Innovation

• Dynamic Graph Isomorphism
• Planar Graphs
• DynFO
• First-Order Logic
• Parallel Algorithm