Lovász (1967) showed that two graphs $G$ and $H$ are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs, i.e., $G$ and $H$ admit the same number of number of homomorphisms from every graph $F$. Subsequently, a substantial line of work studied homomorphism indistinguishability over restricted graph classes. For example, homomorphism indistinguishability over minor-closed graph classes $\mathcal{F}$ such as the class of planar graphs, the class of graphs of treewidth $\leq k$, pathwidth $\leq k$, or treedepth $\leq k$, was shown to be equivalent to quantum isomorphism and equivalences with respect to counting logic fragments, respectively. Via such characterisations, the distinguishing power of e.g. logical or quantum graph isomorphism relaxations can be studied with graph-theoretic means. In this vein, Roberson (2022) conjectured that homomorphism indistinguishability over every graph class excluding some minor is not the same as isomorphism. We prove this conjecture for all vortex-free graph classes. In particular, homomorphism indistinguishability over graphs of bounded Euler genus is not the same as isomorphism. As a negative result, we show that Roberson's conjecture fails when generalised to graph classes excluding a topological minor. Furthermore, we show homomorphism distinguishing closedness for several graph classes including all topological-minor-closed and union-closed classes of forests, and show that homomorphism indistinguishability over graphs of genus $\leq g$ (and other parameters) forms a strict hierarchy.

翻译：Lovász (1967) 证明了两个图 $G$ 和 $H$ 同构当且仅当它们在所有图上具有同态不可区分性，即对于任意图 $F$，从 $F$ 到 $G$ 和 $H$ 的同态数量相同。随后，大量研究工作聚焦于受限图类上的同态不可区分性。例如，在诸如平面图、树宽 $\leq k$ 的图、路宽 $\leq k$ 的图或树深 $\leq k$ 的图等子式封闭图类 $\mathcal{F}$ 上的同态不可区分性，已被证明分别等价于量子同构以及与计数逻辑片段相关的等价关系。通过此类刻画，可以利用图论方法研究逻辑或量子图同构松弛的区分能力。基于这一思路，Roberson (2022) 推测：对于任意排除某一子式的图类，其上的同态不可区分性均不等价于同构。我们针对所有无涡图类证明了该猜想。特别地，在有界欧拉亏格图类上的同态不可区分性不等价于同构。作为否定性结果，我们证明了 Roberson 猜想在推广至排除某一拓扑子式的图类时不成立。此外，我们证明了若干图类（包括所有拓扑子式封闭的森林类及并封闭的森林类）具有同态区分封闭性，并证明了在亏格 $\leq g$（及其他参数）的图类上的同态不可区分性构成严格层级结构。