Logarithmic WeisfeilerLeman Identifies All Planar Graphs
Abstract
The WeisfeilerLeman (WL) algorithm is a wellknown combinatorial procedure for detecting symmetries in graphs and it is widely used in graphisomorphism 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 nonisomorphic graph) by the kdimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3connected planar graphs. The number of iterations needed by the kdimensional 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 firstorder logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}sentence of logarithmic quantifier depth.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.16218
 Bibcode:
 2021arXiv210616218G
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Computer Science  Logic in Computer Science;
 Mathematics  Combinatorics
 EPrint:
 21 pages, 2 figures, accepted at ICALP 2021