Homomorphism counting for immersion-closed classes is not isomorphism

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.

• The paper proves that every proper, union-closed, immersion-closed graph class contains homomorphism tests that fail to distinguish some non-isomorphic graphs, extending results for bounded-degree classes.
• It develops structural oddomorphism results showing that a t-oddomorphism forces a K_t immersion, while a (t+2)-oddomorphism forces treewidth at least t+1, with the treewidth bound tight.
• The results separate immersion closure from topological-minor closure and leave open whether the oddomorphism-to-immersion bound can be reduced to a linear constant, potentially c=1, while Roberson’s minor-closed conjecture remains unresolved.

The paper "Homomorphism counting for immersion-closed classes is not isomorphism" (2602.08738) establishes that for every proper, union-closed class of graphs $\mathcal{M}$ that is closed under taking immersions, there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G) = \hom(K,H)$ for all $K \in \mathcal{M}$. This settles the immersion-analogue of a conjecture of Roberson concerning minor-closed classes and demonstrates that minor-closed classes do not hold exclusive claim to the property that counting homomorphisms from them fails to capture isomorphism. The argument rests on two structural results about oddomorphisms—a notion introduced in prior work—which are themselves of independent combinatorial interest: every graph admitting a $t$-oddomorphism contains a $K_t$-immersion, and every graph admitting a $(t+2)$-oddomorphism has treewidth at least $t+1$, with the latter bound being tight.

Background and motivation

Lovász's classical theorem states that $\hom(K,G) = \hom(K,H)$ for all graphs $G$0 if and only if $G$1 and $G$2 are isomorphic (2602.08738). Restricting the test class $G$3 yields coarser equivalence relations: trees give colour refinement (the 1-dimensional Weisfeiler–Leman algorithm), by a result of Dvořák; graphs of treewidth at most $G$4 give equivalence to indistinguishability by $G$5-WL; cycles give cospectrality; planar graphs yield quantum isomorphism, for which the associated equivalence problem is undecidable.

Several natural classes $G$6 fail to yield isomorphism testing—for instance, Dvořák showed that counting from all 2-degenerate graphs suffices only up to non-isomorphic indistinguishable pairs, and bounded-degree classes also fail. Roberson conjectured that every proper minor-closed and union-closed class admits such indistinguishable pairs. This conjecture gained plausibility from a result of Neuen and Seppelt showing that counting homomorphisms even from the class of $G$7-topological-minor-free graphs does determine isomorphism—so topological-minor-closed classes cannot serve as counterexamples, suggesting minor closure marks a genuine threshold.

Main result

The paper proves:

Theorem. If $G$8 is a proper immersion-closed and union-closed graph class, then there exist non-isomorphic graphs $G$9 such that $H$0 for all $H$1 (2602.08738).

Since every maximum-degree-$H$2 class is immersion-closed but not minor-closed for $H$3, this strictly generalizes Roberson's earlier result for bounded-degree graphs and shows that any "special role" of minor-closed classes must be shared with immersion-closed ones. Notably, the result cannot be extended in the natural direction: replacing immersions by topological minors would contradict Neuen and Seppelt's theorem, since $H$4-topological-minor-free classes are proper and topological-minor-closed yet induce isomorphism. Immersion and minor orders are incomparable, though both are well-quasi-orders by Robertson–Seymour theory, so the result occupies a genuinely distinct position in the hierarchy of closure operators.

Oddomorphisms and structural results

The technical engine is the theory of oddomorphisms. A proper $H$5-colouring $H$6 of $H$7 is a $H$8-oddomorphism if every vertex sees an odd number of neighbours of each foreign colour or an even number of each (i.e., every vertex is $H$9-odd or $\hom(K,G) = \hom(K,H)$0-even), and each colour class contains an odd number of $\hom(K,G) = \hom(K,H)$1-odd vertices; notation $\hom(K,G) = \hom(K,H)$2. Two preservation lemmas underpin the analysis:

• Bi-coloured cycle removal: deleting a cycle within the union of two colour classes preserves the oddomorphism, so minimal examples are acyclic on every pair of colour classes.
• $\hom(K,G) = \hom(K,H)$3-merger: identifying same-coloured vertices $\hom(K,G) = \hom(K,H)$4 via symmetric difference of neighbourhoods while deleting edge-disjoint 2-coloured $\hom(K,G) = \hom(K,H)$5-paths preserves the oddomorphism, with parity bookkeeping showing that merged vertices are $\hom(K,G) = \hom(K,H)$6-odd exactly when exactly one of $\hom(K,G) = \hom(K,H)$7 was $\hom(K,G) = \hom(K,H)$8-odd.

From large oddomorphism to clique immersion

The first structural theorem states that if $\hom(K,G) = \hom(K,H)$9 admits a $K \in \mathcal{M}$0-oddomorphism then $K \in \mathcal{M}$1 contains a $K \in \mathcal{M}$2-immersion (2602.08738). The proof considers a vertex- and edge-minimal counterexample, which must be acyclic on each pair of colour classes. Splitting off paths between $K \in \mathcal{M}$3-odd endpoints across colour pairs produces a graph $K \in \mathcal{M}$4 where either some pair of same-coloured vertices carries at least $K \in \mathcal{M}$5 parallel edges, or the simplified graph has minimum degree at least $K \in \mathcal{M}$6. In the latter case the $K \in \mathcal{M}$7-immersion follows from Gauthier–Le–Wollan's minimum-degree forcing theorem; in the former case, the $K \in \mathcal{M}$8 corresponding 2-coloured paths permit a merger reducing the instance, and any resulting $K \in \mathcal{M}$9-immersion lifts back since each path in the lifted immersion uses the merged vertex at most once and there are enough disjoint paths to accommodate the $t$0 branch pairs.

The authors concede the quantitative aspect is not optimal: they conjecture an absolute constant $t$1 such that a $t$2-oddomorphism forces a $t$3-immersion, possibly even $t$4.

A tight treewidth bound

The second structural result gives a best-possible lower bound: a graph with a $t$5-oddomorphism has treewidth at least $t$6 (2602.08738). Tightness holds because $t$7 via the identity and $t$8. The proof proceeds by contradiction on a minimal counterexample with a width-$t$9 tree decomposition, rooting the decomposition tree and selecting an $K_t$0-odd vertex appearing latest in breadth-first order. Parity constraints force a neighbour whose colour is absent from its bag; a Kempe-chain argument then yields two same-coloured vertices whose merger preserves both the oddomorphism and treewidth, producing a smaller counterexample.

This theorem immediately recovers, by a new route, the known fact that homomorphism counts from bounded-treewidth graphs do not determine isomorphism—though this was already established via the connection between treewidth-$K_t$1 homomorphism indistinguishability and $K_t$2-WL, which is known not to capture isomorphism.

Reduction to the counting statement

The main theorem follows from an equivalent reformulation: for every $K_t$3, there exist non-isomorphic $K_t$4 such that any $K_t$5 with $K_t$6 contains a $K_t$7-immersion (2602.08738). Equivalence with the main theorem is straightforward: given the reformulation, take $K_t$8 excluding $K_t$9; conversely, apply the main theorem to the class of $(t+2)$0-immersion-free graphs.

To prove the reformulation, the authors invoke Roberson's construction producing non-isomorphic $(t+2)$1 with $(t+2)$2, strict inequality holding precisely when $(t+2)$3 admits a weak oddomorphism into a chosen target graph $(t+2)$4. Setting $(t+2)$5 appropriately ensures any distinguishing $(t+2)$6 contains a subgraph with an oddomorphism into $(t+2)$7, hence—by the immersion theorem—a $(t+2)$8-immersion. Non-isomorphism of $(t+2)$9 follows from Lovász's theorem applied via Roberson's Lemma 3.14.

Limitations and open questions

Three gaps are explicitly acknowledged. First, the constant relating oddomorphism size to immersion size is likely far from optimal; the conjectured linear dependence with constant $t+1$0 (and possibly $t+1$1) remains open, with the treewidth theorem confirming $t+1$2 only for $t+1$3. Second, Roberson's original minor-closed conjecture itself is untouched—the paper establishes only the immersion analogue, and the question of whether every proper minor-closed union-closed class fails to induce isomorphism stands. Third, a strengthening conjecturing that distinct immersion-closed union-closed classes induce distinct equivalence relations (i.e., that the map $t+1$4 is injective) is stated without proof.

Conclusion

This paper demonstrates that homomorphism counting over any proper immersion-closed, union-closed class never yields an isomorphism test, extending the bounded-degree case and delineating a sharp boundary against Neuen–Seppelt's positive result for $t+1$5-topological-minor-free classes. The supporting results—that large oddomorphisms force large clique immersions and that $t+1$6-oddomorphisms force treewidth at least $t+1$7, tightly—are substantive contributions connecting parity-based colouring structure to connectivity and decomposition parameters. The remaining question posed by the work is whether the oddomorphism-to-immersion relationship can be made linear in the strong sense ($t+1$8), and whether Roberson's minor-closed conjecture admits a similar reduction to a purely structural forcing statement.