We construct a pair of non-isomorphic, bipartite graphs which are not distinguished by counting the number of homomorphisms to any tree. This answers a question motivated by Atserias et al. (LICS 2021). In order to establish the construction, we analyse the equivalence relations induced by counting homomorphisms to trees of diameter two and three and obtain necessary and sufficient conditions for two graphs to be equivalent. We show that three is the optimal diameter for our construction.

翻译：我们构造了一对非同构的二部图，它们无法通过计算到任意树的同态数量来区分。这回答了由Atserias等人（LICS 2021）提出的问题。为了建立该构造，我们分析了通过计算到直径为二和三的树的同态所诱导的等价关系，并获得了两个图等价的充分必要条件。我们证明直径三对于我们的构造是最优的。