Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$ denotes the number of homomorphisms from $F$ to $G$. A classical result of Lov\'{a}sz shows that isomorphism between graphs is equivalent to homomorphism indistinguishability over the class of all graphs. More recently, there has been a series of works giving natural algebraic and/or logical characterizations for homomorphism indistinguishability over certain restricted graph classes. A class of graphs $\mathcal{F}$ is homomorphism-distinguishing closed if, for every $F \notin \mathcal{F}$, there are graphs $G$ and $H$ such that $G \equiv_{\mathcal{F}} H$ and $\operatorname{hom}(F,G) \neq \operatorname{hom}(F,H)$. Roberson conjectured that every class closed under taking minors and disjoint unions is homomorphism-distinguishing closed which implies that every such class defines a distinct equivalence relation between graphs. In this note, we confirm this conjecture for the classes $\mathcal{T}_k$, $k \geq 1$, containing all graphs of tree-width at most $k$. As an application of this result, we also characterize which subgraph counts are detected by the $k$-dimensional Weisfeiler-Leman algorithm. This answers an open question from [Arvind et al., J. Comput. Syst. Sci., 2020].

翻译：两个图关于图类 $\mathcal{F}$ 是同态不可区分的，记为 $G \equiv_{\mathcal{F}} H$，如果对所有 $F \in \mathcal{F}$，均有 $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$，其中 $\operatorname{hom}(F,G)$ 表示从 $F$ 到 $G$ 的同态个数。Lovász 的一个经典结果表明，图之间的同构等价于关于所有图类的同态不可区分性。近年来，一系列工作给出了关于某些受限图类的同态不可区分性的自然代数与/或逻辑刻画。一个图类 $\mathcal{F}$ 是**同态区分封闭的**，如果对每个 $F \notin \mathcal{F}$，存在图 $G$ 和 $H$，使得 $G \equiv_{\mathcal{F}} H$ 但 $\operatorname{hom}(F,G) \neq \operatorname{hom}(F,H)$。Roberson 猜想：每个在取子式和不相交并下封闭的图类都是同态区分封闭的，这意味着每个这样的类都在图之间定义了一个不同的等价关系。在本文中，我们证实了对于类 $\mathcal{T}_k$（$k \geq 1$）的猜想成立，其中 $\mathcal{T}_k$ 包含所有树宽至多为 $k$ 的图。作为该结果的一个应用，我们还刻画了哪些子图计数可以被 $k$ 维 Weisfeiler-Leman 算法检测到。这回答了 [Arvind 等, J. Comput. Syst. Sci., 2020] 中的一个开放问题。