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

TL;DR

This paper establishes tight lower bounds on the depth needed for universal covers and color refinement to distinguish graphs, showing that these bounds are linear in the number of vertices, which answers a longstanding question in graph isomorphism theory.

Contribution

The paper proves that the minimal depth for universal covers to distinguish graphs is asymptotically equal to the number of vertices, providing the first linear lower bound and answering Norris's question.

Findings

Universal cover depth T(n) is asymptotically (2-o(1))n.

Color refinement requires (1-o(1))n rounds for certain graphs.

Disjoint union of graphs requires about 2n rounds for color stabilization.

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.