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 $k$.

翻译：两个图关于图类 $\mathcal{F}$ 同态不可区分，记为 $G \equiv_{\mathcal{F}} H$，若对所有 $F \in \mathcal{F}$ 有 $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$，其中 $\operatorname{hom}(F,G)$ 表示从 $F$ 到 $G$ 的同态数量。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。近期，一系列工作给出了关于特定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 $\mathcal{F}$ 称为同态区分闭的，若对每个 $F \notin \mathcal{F}$，存在图 $G$ 和 $H$ 使得 $G \equiv_{\mathcal{F}} H$ 且 $\operatorname{hom}(F,G) \neq \operatorname{hom}(F,H)$。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闭的，这意味着每个此类图类在图上定义了一个不同的等价关系。在本文中，我们证实了对于类 $\mathcal{T}_k$（$k \geq 1$，包含所有树宽至多为 $k$ 的图）该猜想成立。