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 $G$ equals 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. In this article, the interplay of the properties of a graph class and its homomorphism indistinguishability relation are studied. As an application, self-complementarity, a property of logics on graphs satisfied by many well-studied logics, is identified. It is proven that the equivalence over a self-complementary logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Thereby, first evidences are provided for a possible connection between minors and homomorphism indistinguishability as conjectured by Roberson (2022).

翻译：图$G$和$H$在图类$\mathcal{F}$上称为同态不可区分，若对所有图$F \in \mathcal{F}$，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，例如（量子）同构、谱等价和逻辑等价，均可描述为特定图类上的同态不可区分关系。本文研究了图类性质与其同态不可区分关系之间的相互作用。作为应用，我们识别出图的逻辑性质中的自互补性，该性质被许多得到充分研究的逻辑所满足。我们证明，若一个自互补逻辑可刻画为同态不可区分关系，则其等价关系可由某个对子图封闭的图类上的同态不可区分关系所刻画。由此，我们为Roberson（2022）所猜测的子图与同态不可区分性之间的可能联系提供了初步证据。