Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

As it is well known, 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-F\"urer-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 much 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 graphs of color multiplicity 4 with directed and 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-F\"urer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-F\"urer-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.