Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs

Abstract: Lovász (1967) showed that two graphs $G$ and $H$ are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs, i.e., $G$ and $H$ admit the same number of number of homomorphisms from every graph $F$. Subsequently, a substantial line of work studied homomorphism indistinguishability over restricted graph classes. For example, homomorphism indistinguishability over minor-closed graph classes $\mathcal{F}$ such as the class of planar graphs, the class of graphs of treewidth $\leq k$, pathwidth $\leq k$, or treedepth $\leq k$, was shown to be equivalent to quantum isomorphism and equivalences with respect to counting logic fragments, respectively. Via such characterisations, the distinguishing power of e.g. logical or quantum graph isomorphism relaxations can be studied with graph-theoretic means. In this vein, Roberson (2022) conjectured that homomorphism indistinguishability over every graph class excluding some minor is not the same as isomorphism. We prove this conjecture for all vortex-free graph classes. In particular, homomorphism indistinguishability over graphs of bounded Euler genus is not the same as isomorphism. As a negative result, we show that Roberson's conjecture fails when generalised to graph classes excluding a topological minor. Furthermore, we show homomorphism distinguishing closedness for several graph classes including all topological-minor-closed and union-closed classes of forests, and show that homomorphism indistinguishability over graphs of genus $\leq g$ (and other parameters) forms a strict hierarchy.

• **The paper demonstrates the weak Roberson conjecture for vortex-free minor-closed classes:** For every $k extgreater{} 0$, non-isomorphic graphs exist that are homomorphism indistinguishable over all graphs of Hadwiger number at most $k$, where $\mathcal{H}_k$ is the class of graphs of vortex-free Hadwiger number at most $k$ [Thilikos et al ].
• **The paper disproves the generalisation of the conjecture beyond excluded minors,** inferring that the intersection of homomorphism indistinguishability over all graphs and isomorphism occurs at $K_5$.
• **The paper presents techniques to verify homomorphism distinguishing closedness without appealing to characterisations,** such as the deletion and elimination distance, separator lemma, and clique-sums. This includes applications in Wagner decompositions and new classifications like self-contained proofs that treedepth-$k$ graphs are homomorphism distinguishing closed, previously established via Ehrenfeucht–Fraïssé games.

Background and motivation

Lovász's classical theorem states that two graphs are isomorphic exactly when they admit the same number of homomorphisms from every graph (2601.18602). A substantial line of work refines this by restricting the test graphs: homomorphism indistinguishability over graphs of treewidth at most $k$ corresponds to indistinguishability in the counting logic fragment $\mathsf{C}^{k+1}$ and the $k$-dimensional Weisfeiler–Leman algorithm, while indistinguishability over planar graphs corresponds to quantum isomorphism. The paper studies how sparsity properties of a graph class $\mathcal{F}$ govern the distinguishing power of the relation $\equiv_{\mathcal{F}}$, addressing two conjectures of Roberson.

The weak Roberson conjecture asserts that for every $k$, there exist non-isomorphic graphs that are homomorphism indistinguishable over all graphs of Hadwiger number at most $k$ — equivalently, over every graph class excluding some minor. Prior to this work, only the cases of bounded treewidth (via Cai–Fürer–Immerman) and planar graphs (via quantum isomorphism) were known; even the case $k=5$ rested on an unpublished result of Richter, Roberson, and Thomassen.

The strong Roberson conjecture asserts that every minor-closed and union-closed graph class is homomorphism distinguishing closed, meaning that adding any single excluded graph $F'$ to $\mathcal{F}$ strictly refines $\mathsf{C}^{k+1}$0. This property is what allows homomorphism indistinguishability relations to be compared purely by inclusion of their defining classes, and it underpins applications in finite model theory, semidefinite optimisation hierarchies, and graph neural network expressivity.

Separation from isomorphism for vortex-free minor-closed classes

The first main theorem proves the weak Roberson conjecture for all minor-closed classes omitting vortices: for every $\mathsf{C}^{k+1}$1 there exist non-isomorphic $\mathsf{C}^{k+1}$2 with $\mathsf{C}^{k+1}$3, where $\mathsf{C}^{k+1}$4 is the class of graphs of vortex-free Hadwiger number at most $\mathsf{C}^{k+1}$5, as characterised by Thilikos et al. Notable special cases include all graphs embeddable on a surface of Euler genus at most $\mathsf{C}^{k+1}$6 (orientable or not) and all classes excluding a single-crossing minor. This substantially extends the previously known instances, though it stops short of the full conjecture because the Robertson–Seymour structure theorem's vortex ingredient remains unhandled.

In contrast, the paper shows the conjecture does not generalise beyond excluded minors. Two graphs are isomorphic if and only if they are homomorphism indistinguishable over all graphs excluding $\mathsf{C}^{k+1}$7 as a topological minor. The same holds for classes of bounded $\mathsf{C}^{k+1}$8-admissibility and bounded local treewidth. Since $\mathsf{C}^{k+1}$9-topological-minor-free graphs have treewidth at most 2, the threshold $k$0 is optimal. This delineates a sharp demarcation: minor-closedness appears to be precisely the property that permits non-isomorphism, whereas topological-minor exclusion forces full distinguishing power.

The mechanism behind the negative result is a lemma showing that if $k$1 is closed under subgraphs and 2-sums with triangles, then its homomorphism distinguishing closure $k$2 is minor-closed. The proof uses Lovász–Szegedy series-parallel contractors to simulate edge contractions via homomorphism counts from triangle-augmented subgraphs of members of $k$3. Because every graph is a minor of a subcubic graph, and subcubic graphs exclude $k$4 topologically, the class obtained from subcubic graphs by iterated 2-sums with triangles has trivial distinguishing closure — hence isomorphism.

Homomorphism distinguishing closedness

On the positive side of the strong conjecture, the paper resolves it for all union-closed classes of forests closed under topological minors, generalising prior results for forests of bounded maximum degree. It also establishes closedness for cactus graphs, outerplanar graphs, $k$5-minor-free graphs, $k$6-minor-free graphs, disjoint unions of $k$7-apex planar graphs, and disjoint unions of graphs of vertex cover or feedback vertex set number at most $k$8. Several of these reproduce earlier results (planarity aside) by purely combinatorial arguments rather than through logical or algebraic characterisations — an important methodological advance given that most minor-closed classes are unlikely to admit usable characterisations of $k$9.

A concrete consequence answers a question of Roberson regarding Lasserre relaxations: there exist graphs whose first-level Lasserre relaxation of the graph isomorphism integer program is feasible yet which are distinguished by 2-WL. This follows because outerplanar graphs are homomorphism distinguishing closed while $\mathcal{F}$0 has treewidth 2 but is not outerplanar.

Techniques: oddomorphisms, separators, and clique-sums

The unifying technical framework is Roberson's criterion via CFI graphs: a graph $\mathcal{F}$1 distinguishes the even and odd CFI graphs over a connected base graph $\mathcal{F}$2 exactly when there exists a weak oddomorphism $\mathcal{F}$3 — a homomorphism satisfying parity constraints on fibres. Consequently, a componental class closed under weak oddomorphisms is homomorphism distinguishing closed. The paper develops combinatorial machinery to verify such closure without appealing to characterisations:

• Deletion and elimination distance: if $\mathcal{F}$4 is closed under weak oddomorphisms and subgraphs, so are the classes of bounded deletion distance and bounded elimination distance to $\mathcal{F}$5. The latter yields a self-contained proof that treedepth-$\mathcal{F}$6 graphs are homomorphism distinguishing closed, previously established via Ehrenfeucht–Fraïssé games.
• Separator lemma: for an oddomorphism $\mathcal{F}$7 where $\mathcal{F}$8 has more components than $\mathcal{F}$9, a linear-algebraic argument over $\equiv_{\mathcal{F}}$0 selects a proper subset of components of $\equiv_{\mathcal{F}}$1 whose union, together with a clique-compressed image of $\equiv_{\mathcal{F}}$2, forms a minor admitting an oddomorphism to $\equiv_{\mathcal{F}}$3. This drives the clique-sum closure results.
• Clique-sums: minimal forbidden subgraphs of $\equiv_{\mathcal{F}}$4 are 2-connected, and minimal forbidden minors of $\equiv_{\mathcal{F}}$5 (for minor-closed $\equiv_{\mathcal{F}}$6) are 3-connected; combined with the separator lemma, closure under weak oddomorphisms lifts from $\equiv_{\mathcal{F}}$7 to $\equiv_{\mathcal{F}}$8 when $\equiv_{\mathcal{F}}$9 is $k$0-connected. Applications include Wagner's decompositions of $k$1- and $k$2-minor-free graphs, the latter requiring an exhaustive check that no minor of the Wagner graph $k$3 admits an oddomorphism to $k$4.
• Bounded-treewidth arguments: a structural lemma producing low-degree non-cut vertices in graphs of small treewidth, plus twin-reduction lemmas, yields the key step that any treewidth-2 graph admitting an oddomorphism to $k$5 contains $k$6 as a minor — whence outerplanar closedness.

Strict hierarchies

Although closedness of the genus classes $k$7 and the vortex-free Hadwiger classes $k$8 remains open, the paper proves both form strict infinite hierarchies: for every $k$9 there exist $k$0 but $k$1, and analogously for $k$2. The separating witnesses are built from disjoint unions of $k$3's augmented with apex vertices, exploiting that genus-$k$4 graphs exclude $k$5-Kuratowski minors (Battle–Harary–Kodama–Youngs) and that the classes $k$6 excluding $k$7-Kuratowski minors inherit oddomorphism closure from planar graphs. These give two new provably infinite chains of graph isomorphism relaxations, each strictly between 1-WL-type and full isomorphism.

Limitations and open questions

The central limitation is that the weak Roberson conjecture remains open in full: the role of vortices in the Robertson–Seymour structure theorem is unresolved, and the paper explicitly asks whether homomorphism indistinguishability over planar graphs with one vortex equals isomorphism. Progress here appears blocked by the fact that the only known proof that oddomorphisms preserve planarity relies on the quantum-isomorphism characterisation; a purely graph-theoretic proof is identified as a desideratum. Similarly, homomorphism distinguishing closedness of $k$8 and $k$9 is not established — only strict separation of successive levels. For dense graph classes, nothing in the paper addresses whether $k=5$0 over cliquewidth-bounded graphs can be separated from isomorphism. Finally, the computational complexity of deciding $k=5$1 for fixed proper minor-closed $k=5$2 of unbounded treewidth is conjectured undecidable but known only for planar graphs.

Conclusion

This paper advances the theory of homomorphism indistinguishability on three fronts. It confirms Roberson's weak conjecture for all vortex-free minor-closed classes, establishing in particular that bounded-genus homomorphism counts never capture isomorphism, while proving that the conjecture fails for topological-minor exclusions and related sparsity notions — thereby identifying minor-closedness as the operative dividing line. It resolves the strong conjecture for all topological-minor-closed union-closed forest classes and adds numerous new closed classes via a self-contained combinatorial toolkit built on oddomorphisms, separator lemmas, and Lovász–Szegedy contractors. And it constructs two strict infinite hierarchies of isomorphism relaxations indexed by Euler genus and vortex-free Hadwiger number. The remaining gap — vortices — now constitutes the precise obstacle to settling the weak Roberson conjecture in full.