Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs

TL;DR

This paper investigates the power of counting homomorphisms from sparse graphs to distinguish non-isomorphic graphs, proving conjectures for certain classes and establishing hierarchies and limitations of this method.

Contribution

It proves Roberson's conjecture for vortex-free classes, shows the limits of homomorphism indistinguishability over minor-excluding classes, and establishes hierarchies among graph classes based on genus.

Findings

Homomorphism indistinguishability over graphs of bounded Euler genus does not imply isomorphism.

Roberson's conjecture holds for vortex-free graph classes.

A strict hierarchy exists among graph classes based on genus and other parameters.

Abstract

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs, i.e., $G$ and $H$ admit the same number of number of homomorphisms from every graph $F$. Subsequently, a substantial line of work studied homomorphism indistinguishability over restricted graph classes. For example, homomorphism indistinguishability over minor-closed graph classes $\mathcal{F}$ such as the class of planar graphs, the class of graphs of treewidth $\leq k$, pathwidth $\leq k$, or treedepth $\leq k$, was shown to be equivalent to quantum isomorphism and equivalences with respect to counting logic fragments, respectively. Via such characterisations, the distinguishing power of e.g. logical or quantum graph isomorphism relaxations can be studied with graph-theoretic means. In this vein, Roberson (2022) conjectured that homomorphism indistinguishability over every graph class excluding some minor is not the same as isomorphism. We prove this conjecture for all vortex-free graph classes. In particular, homomorphism indistinguishability over graphs of bounded Euler genus is not the same as isomorphism. As a negative result, we show that Roberson's conjecture fails when generalised to graph classes excluding a topological minor. Furthermore, we show homomorphism distinguishing closedness for several graph classes including all topological-minor-closed and union-closed classes of forests, and show that homomorphism indistinguishability over graphs of genus $\leq g$ (and other parameters) forms a strict hierarchy.