Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

TL;DR

This paper explores how various graph equivalences can be characterized through homomorphism indistinguishability over minor-closed classes, linking logical, spectral, and quantum graph properties.

Contribution

It establishes that self-complementarity logics correspond to homomorphism indistinguishability over minor-closed classes and classifies finite classes with maximal homomorphism distinguishing properties.

Findings

Homomorphism indistinguishability characterizes many graph equivalences.

Self-complementarity logics relate to minor-closed graph classes.

Classification of finite homomorphism distinguishing closed classes.

Abstract

Two graphs $G$ and $H$ are homomorphism indistinguishable over a class of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ 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.