Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs

TL;DR

This work investigates when two graphs are indistinguishable by homomorphism counts from a restricted graph class $\mathcal{F}$, i.e., $G \equiv_{\mathcal{F}} H$, and how this relates to isomorphism. It develops a robust combinatorial toolkit based on oddomorphisms and separator/clique-sum arguments to prove homomorphism distinguishing closedness for a wide range of sparse classes, and proves a pair of infinite hierarchies (genus and vortex-free Hadwiger number) with strict refinements, separating these from isomorphism. The authors establish the weak Roberson conjecture for all vortex-free minor-closed classes (e.g., bounded-genus graphs) while showing negative results beyond minor exclusions, such as exclusions of a topological minor. They also provide structural results for deletion/elimination distances and outerplanar/forest families, connecting to practical implications for graph neural networks and counting-based isomorphism relaxations. Overall, the paper clarifies the landscape of sparsity-based homomorphism indistinguishability, highlighting both the power and limits of these relaxations for isomorphism testing and related computational questions.

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.

Figures (4)

• Figure 1: Properties of a graph class $\mathcal{F}$ that do (green) or do not (red) guarantee that there exist non-isomorphic graphs $G$ and $H$ that are homomorphism indistinguishable over $\mathcal{F}$.
• Figure 2: An oddomorphism () to the $5$-vertex path. Even vertices are depicted as and odd vertices as .
• Figure 3: The Wagner graph $V_8$.
• Figure 4: Bilabelled graphs used in the proof of \ref{['lem:contractors-for-minors']} in wire notation of mancinska_quantum_2020. The gray wires to the left/right indicate the position of the first/second label. Actual edges are depicted in black.

Theorems & Definitions (106)

• Conjecture 1.1: roberson_oddomorphisms_2022
• Theorem 1.2: label=thm:main-vortex-free-hadwiger
• Theorem 1.3: restate=thmKfive,label=thm:k5-top-minor
• Conjecture 1.4: roberson_oddomorphisms_2022
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8: restate=thmGenusStrictlyRefining,label=thm:genus-strictly-refining
• Theorem 1.9: restate=thmHadStrictlyRefining,label=thm:had-strictly-refining
• Lemma 1.10