Paper Info
Title
Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm.
Abstract
It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fiirer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fiirer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fiirer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n(3))-configurations in incidence geometry.
Year
DOI
Venue
2020
10.4230/LIPIcs.STACS.2020.43
Leibniz International Proceedings in Informatics
Keywords
DocType
Volume
Graph Isomorphism,Weisfeiler-Leman Algorithm,Cai-Furer-Immerman Graphs,coherent Configurations
Conference
154
ISSN
Citations 
PageRank 
1868-8969
0
0.34
References 
Authors
0
3
Name
Order
Citations
PageRank
Frank Fuhlbrück102.03
Johannes Köbler258046.51
Oleg Verbitsky319127.50