Logarithmic Weisfeiler-Leman Identifies 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.

On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)

“…For instance, Weisfeiler-Leman is a key subroutine in Babai's quasipolynomial-time GI algorithm [4]. Furthermore, Weisfeiler-Leman has led to advances in simultaneously developing both efficient isomorphism tests and the descriptive complexity theory for finite graphs-see for instance, [32,38,49,50,34,36,35,51,1,2,64]. Weisfeiler-Leman also has close connections to the Sherali-Adams hierarchy in linear programming [37].…”