Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs

2026-01-26

AI Summary

This work investigates whether homomorphism indistinguishability over sparse graph classes—such as vortex-free graphs and graphs of bounded Euler genus—is equivalent to graph isomorphism. By integrating techniques from homomorphism counting, minor and topological subgraph theory, and structural graph parameters including treewidth, treedepth, and genus, the study analyzes the distinguishing power of logical and quantum relaxations of isomorphism. The main contributions include confirming Roberson’s conjecture for vortex-free graph classes, demonstrating that homomorphism indistinguishability fails to capture isomorphism on bounded Euler genus graphs and breaks down when excluding topological subgraphs, establishing closure properties of homomorphism distinguishability across several graph classes, and constructing a parameterized strict hierarchy reflecting these distinctions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Lov\'asz (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.

Problem

• graph isomorphism
• homomorphism indistinguishability
• minor-closed classes
• topological minor
• graph distinguishability

Innovation

• homomorphism indistinguishability
• graph minors
• quantum isomorphism
• sparse graph classes
• Euler genus