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 question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.

翻译：两个图$G$和$H$关于图类$\mathcal{F}$同态不可区分，若对所有$F \in \mathcal{F}$，从$F$到$G$的同态数量等于从$F$到$H$的同态数量。许多比较图的自然等价关系，如（量子）同构、谱等价和逻辑等价，可表征为特定图类上的同态不可区分关系。从这些丰富实例中抽象，本文证明：任何满足可表征为同态不可区分关系的自补逻辑的等价关系，均可由某个子式封闭图类上的同态不可区分关系表征。自补性是大多数被充分研究的逻辑满足的温和性质。该结果源于图类的闭包性质与其同态不可区分关系的保持性质之间的对应关系。此外，我们分类了所有在某种意义下有限（本质上是近似有限）且满足同态区分封闭的最大性条件（即向类中添加任何图都会严格细化其同态不可区分关系）的图类。由此，我们解答了Roberson (2022)关于同态区分闭包一般性质提出的若干问题。