TL;DR
This paper constructs non-isomorphic bipartite graphs indistinguishable by homomorphism counts to any tree, answering a question about the limitations of tree homomorphism-based graph invariants.
Contribution
It provides the first explicit construction of such graphs and characterizes the equivalence relations induced by homomorphism counts to small-diameter trees.
Findings
Constructed bipartite graphs indistinguishable by tree homomorphism counts
Analyzed equivalence relations for trees of diameter two and three
Established diameter three as the minimal for such constructions
Abstract
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.
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