Universal covers, color refinement, and two-variable counting logic: Lower bounds for the depth

Abstract

Given a connected graph $G$ and its vertex $x$, let $U_x(G)$ denote the universal cover of $G$ obtained by unfolding $G$ into a tree starting from $x$. Let $T=T(n)$ be the minimum number such that, for graphs $G$ and $H$ with at most $n$ vertices each, the isomorphism of $U_x(G)$ and $U_y(H)$ surely follows from the isomorphism of these rooted trees truncated at depth $T$. Motivated by applications in theory of distributed computing, Norris [Discrete Appl. Math. 1995] asks if $T(n)\le n$. We answer this question in the negative by establishing that $T(n)=(2-o(1))n$. Our solution uses basic tools of finite model theory such as a bisimulation version of the Immerman-Lander 2-pebble counting game. The graphs $G_n$ and $H_n$ we construct to prove the lower bound for $T(n)$ also show some other tight lower bounds. Both having $n$ vertices, $G_n$ and $H_n$ can be distinguished in 2-variable counting logic only with quantifier depth $(1-o(1))n$. It follows that color refinement, the classical procedure used in isomorphism testing and other areas for computing the coarsest equitable partition of a graph, needs $(1-o(1))n$ rounds to achieve color stabilization on each of $G_n$ and $H_n$. Somewhat surprisingly, this number of rounds is not enough for color stabilization on the disjoint union of $G_n$ and $H_n$, where $(2-o(1))n$ rounds are needed.