Dynamic Planar Graph Isomorphism is in DynFO

TL;DR

This paper demonstrates that the problem of maintaining isomorphism between two dynamic planar graphs can be efficiently managed using first-order logic, placing it within the DynFO complexity class.

Contribution

It introduces a method to maintain dynamic planar graph isomorphism using first-order logic and polynomial auxiliary data, a novel result in dynamic descriptive complexity.

Findings

Maintains graph isomorphism with first-order logic formulas.

Places the dynamic planar graph isomorphism problem in DynFO.

Enables a constant-time parallel algorithm for the problem.

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.