Weisfeiler-lehman refinement requires at least a linear number of iterations

Abstract

Let Lk,m be the set of formulas of first order logic containing only variables from x1, x2, ... xk and having quantifier depth at most m. Let Ck,m be the extension of L k,m obtained by allowing counting quantifiers meaning that there are at least i distinct xj 's. It is shown that for constants h ≥ 1, there are pairs of graphs such that h-dimensional Weisfeiler-Lehman refinement (h-dim W-L) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2h-dim W-L only requires O(n) iterations, and 3h-1-dim W-L only requires O(log n) iterations. In terms of logic, this means that there is a c > 0 and a class of non-isomorphic pairs (GhnHhn) of graphs with G hn and Hhn having O(n) vertices such that the same sentences of Lh+1cn and Ch+1cn hold (h + 1 variables, depth cn), even though Ghn and H hn can already be distinguished by a sentence of L k,m and thus Ckm for some k > h and m = O(log n). © 2011 Springer-Verlag Berlin Heidelberg.