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.

翻译：图G和H被称为在给定图类F上的同态不可区分，若对所有图F∈F，从F到G的同态数量等于从F到H的同态数量。许多自然定义的图等价关系（如（量子）同构、谱等价和逻辑等价）均可表征为特定图类上的同态不可区分关系。本文从这些丰富的实例中抽象出一般规律，证明任何满足"自互补性"（即具有同态不可区分关系表征性的逻辑）的等价关系，均可通过某个子闭图类上的同态不可区分性来刻画。自互补性是一个温和的性质，为大多数被广泛研究的逻辑所满足。该结果源于图类封闭性质与其同态不可区分关系保持性质之间的对应。此外，我们分类了所有在某种意义上有限（本质上是拟有限）且满足极大性条件（即同态区分封闭）的图类——向该图类添加任何图都会严格细化其同态不可区分关系。由此，我们回答了Roberson（2022）关于同态区分闭包一般性质的若干问题。