文章基本信息

标题：Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm
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作者：Frank Fuhlbr{"u}ck ; Johannes K{"o}bler ; Oleg Verbitsky 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：154
页码：43:1-43:18
DOI：10.4230/LIPIcs.STACS.2020.43
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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-FÃ¼rer-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-FÃ¼rer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-FÃ¼rer-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 (nâ,f)-configurations in incidence geometry.
关键词：Graph Isomorphism; Weisfeiler-Leman Algorithm; Cai-FÃ¼rer-Immerman Graphs; coherent Configurations