Homomorphism Counts to Trees
Anuj Dawar
TL;DR
The paper resolves a question about right-$ ext{T}$-equivalence (homomorphism counts into trees) by constructing a pair of bipartite, non-isomorphic graphs $G$ and $H$ with identical counts $| ext{Hom}(G,T)|=| ext{Hom}(H,T)|$ for every tree $T$. It develops a taxonomy of equivalences for trees of diameter $2$ and $3$, introducing the notions of size parameters, partial-sum diffs, and neighbourhood-size equivalence to characterize when two graphs are indistinguishable by tree-homomorphism counts. A key contribution is a diameter-$3$ counterexample built from a permutation-based family $G_ ext{π}$, showing that non-conjugate permutations with the same fixed-point count yield non-isomorphic yet right-equivalent graphs. The explicit pair for $n=4$ demonstrates that the separation is real and optimal at diameter $3$. The work opens questions on higher-diameter trees and seeks a complete characterization of right-equivalence to all trees.
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.