On the Weisfeiler algorithm of depth-$1$ stabilization

Abstract: An origin of the multidimensional Weisfeiler-Leman algorithm goes back to a refinement procedure of deep stabilization, introduced by B. Weisfeiler in a paper included in the collective monograph ``On construction and identification of graphs"(1976). This procedure is recursive and the recursion starts from an algorithm of depth-$1$ stabilization, which has never been discussed in the literature. A goal of the present paper is to show that a simplified algorithm of the depth-$1$ stabilization has the same power as the $3$-dimensional Weisfeiler-Leman algorithm. It is proved that the class of coherent configurations obtained at the output of this simplified algorithm coincides with the class introduced earlier by the third author. As an application we also prove that if there exist at least two nonisomorphic projective planes of order $q$, then the Weisfeiler-Leman dimension of the incidence graph of any projective plane of order $q$ is at least $4$.