The Weisfeiler-Leman Dimension of Planar Graphs is at most 3

TL;DR

This paper proves that the Weisfeiler-Leman dimension of all finite planar graphs is at most 3, significantly improving previous bounds and showing that such graphs are definable with at most 4 variables in first-order logic with counting.

Contribution

The authors establish a tight upper bound of 3 on the WL-dimension of planar graphs, improving previous bounds of 14 and 15, and provide new insights into graph definability and automorphism orbit determination.

Findings

The WL-algorithm correctly tests isomorphism for graphs in minor-closed classes when it determines automorphism orbits.

For most 3-connected planar graphs, individualization plus 1-dimensional WL yields discrete partitions.

Classification of 3-connected planar graphs with fixing number 3.

Abstract

We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best known upper bounds for the dimension and number of variables were 14 and 15, respectively. First we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of any arc-colored 3-connected graph belonging to this class. Then we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two correctly chosen vertices of a colored 3-connected planar graph followed by the 1-dimensional WL-algorithm produces the discrete vertex partition. This implies that the 3-dimensional WL-algorithm determines the orbits of a colored 3-connected planar graph. As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.