Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

TL;DR

This paper proves that the k-dimensional Weisfeiler-Leman algorithm can identify all planar graphs within a logarithmic number of iterations, extending previous results and linking graph identification to logical definability.

Contribution

It establishes a universal logarithmic bound on the WL algorithm's iterations for all planar graphs, generalizing prior work on 3-connected planar graphs.

Findings

WL algorithm identifies all planar graphs within logarithmic iterations

Every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth

Extends previous results from 3-connected to all planar graphs

Abstract

The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm. We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs. The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth.