Homomorphism counting for immersion-closed classes is not isomorphism
Andrea Jiménez, Benjamin Moore, Daniel A. Quiroz, Youngho Yoo
TL;DR
The article addresses when homomorphism counts from a graph class determine isomorphism by proving an immersion-closed and union-closed class $\mathcal{M}$ cannot distinguish some non-isomorphic graphs $G$ and $H$ via all counts $\hom(K,\cdot)$ with $K\in \mathcal{M}$. It develops an oddomorphism framework and proves that a sufficiently large oddomorphism (in terms of $t$) forces a $K_t$-immersion, using a minimal-counterexample strategy and a detailed decomposition of colour classes to lift immersions. This extends Roberson's result for graphs of bounded degree and reinforces the view that immersion-closed classes have a distinctive role in homomorphism indistinguishability, while Neuen and Seppelt show limits for topological-minor closures. Overall, the work broadens the landscape of classes for which homomorphism counts fail to certify isomorphism and highlights the boundary between immersion-based indistinguishability and more restrictive notions.
Abstract
Lovász proved that two graphs $G$ and $H$ are isomorphic if $\hom(K,G) = \hom(K,H)$ for all graphs $K$, where $\hom(G_1,G_2)$ denotes the number of homomorphisms from $G_1$ to $G_2$. Dvořák showed that it suffices to count homomorphisms from all $2$-degenerate graphs $K$. On the other hand, for several interesting graph classes $\mathcal{M}$, it has been shown that there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G)=\hom(K,H)$ for all $K\in \mathcal{M}$. Most such classes are minor-closed classes and Roberson conjectured that every proper minor-closed and union-closed graph class $\mathcal{M}$ has the property of there existing non-isomorphic graphs that are indistinguishable by homomorphism counts from $\mathcal{M}$. There has been an effort to prove Roberson's conjecture as it is believed that minor-closed classes play a special role in the context of homomorphism indistinguishability. We show that this special role, if so, must be shared, by proving an analogue of Roberson's conjecture holds for a rich family of non-minor-closed classes. Namely, we prove that for any proper immersion-closed and union-closed class $\mathcal{M}$, there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G) = \hom(K,H)$ for all $K \in \mathcal{M}$. This extends a result of Roberson on homomorphism indistinguishability over bounded degree graphs, and gives an almost full picture since our result cannot be extended in the natural way, that is, by replacing immersions with topological minors, due to a result of Neuen and Seppelt.