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-Fürer-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-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-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 (n₃)-configurations in incidence geometry.
@InProceedings{fuhlbruck_et_al:LIPIcs.STACS.2020.43, author = {Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg}, title = {{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.43}, URN = {urn:nbn:de:0030-drops-119046}, doi = {10.4230/LIPIcs.STACS.2020.43}, annote = {Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F\"{u}rer-Immerman Graphs, coherent Configurations} }
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