# Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

arXiv: https://arxiv.org/abs/2302.11290 · HTML: https://mlsanity.com/paper/2302.11290/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2023.

## Built on by (most cited first, 3 of 3)

- [Beyond Weisfeiler-Lehman: A Quantitative Framework for GNN Expressiveness](https://mlsanity.com/paper/2401.08514/) (2024)
- [On the Power of the Weisfeiler-Leman Test for Graph Motif Parameters](https://mlsanity.com/paper/2309.17053/) (2023)
- [Homomorphism Expressivity of Spectral Invariant Graph Neural Networks](https://mlsanity.com/paper/2503.00485/) (2025)

## Abstract (arXiv)

Two graphs G and H are homomorphism indistinguishable over a class of graphs F if for all graphs F ∈ F the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.