Planar Graphs: Logical Complexity and Parallel Isomorphism Tests

TL;DR

This paper establishes logical definability and computational complexity bounds for triconnected planar graphs, showing they can be characterized with limited variables and depth, leading to efficient isomorphism testing.

Contribution

It proves that triconnected planar graphs are definable with a fixed number of variables and quantifier depth, enabling $AC^1$ algorithms for isomorphism testing.

Findings

Triconnected planar graphs are definable with at most 15 variables.

A canonical form for these graphs can be computed in $AC^1$.

Planar graph isomorphism is solvable in $AC^1$.

Abstract

We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most $11\log_2 n+43$. As a consequence, a canonic form of such graphs is computable in $AC^1$ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in $AC^1$.