A Log-space Algorithm for Canonization of Planar Graphs

TL;DR

This paper presents a log-space algorithm for canonizing planar graphs, matching the known hardness bounds and improving previous upper bounds, by leveraging graph decompositions and advanced group theoretic techniques.

Contribution

It introduces a log-space algorithm for planar graph canonization, extending Lindell's tree canonization method with new decomposition and group-theoretic insights.

Findings

Planar graph canonization is in logspace.

The algorithm matches the known logspace hardness bounds.

It improves previous AC1 upper bound for planar graph isomorphism.

Abstract

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.