Topological and Geometric Perspectives on Homomorphism Indistinguishability

2026-10-06

AI Summary

This study addresses the long-standing reliance of homomorphism indistinguishability relations on traditional combinatorial methods, which lack a systematic axiomatic characterization. To overcome this limitation, this work introduces topological and geometric perspectives for the first time, integrating algebraic graph theory with Lovász’s theorem to uncover the deep structural properties underlying homomorphism indistinguishability. Furthermore, it proposes a novel characterization of star graph homomorphism parameters. The primary contribution lies in the successful construction of an axiomatic framework for these relations, transcending the constraints of existing combinatorial approaches. Additionally, this research provides new theoretical support for the Ulam–Kelly graph reconstruction conjecture, thereby advancing both the foundational understanding and broader applicability of homomorphism indistinguishability in graph theory.

Abstract

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if, for every graph $F \in \mathcal{F}$, the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Lovász (Acta Mathematica Academiae Scientiarum Hungarica, 1967) showed that two graphs are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs. Subsequently, homomorphism indistinguishability relations of a long list of natural graph classes have been equated with natural graph isomorphism relaxations. Given the wealth of such results, Atserias, Kolaitis, & Wu (LICS 2021) asked for an axiomatic characterisation of homomorphism indistinguishability relations. By exhibiting topological and geometric structure associated with homomorphism indistinguishability, we derive such an axiomatic characterisation. Here, a central ingredient is a novel characterisation of graph parameters of the form $\hom(F, \star)$ for some graph $F$ alternative to a previous result of Lovász & Schrijver (JCTA 2010). Moreover, we investigate the topology of homomorphism indistinguishability and discuss repercussions for the Ulam--Kelly Reconstruction Conjecture.

Problem

• homomorphism indistinguishability
• axiomatic characterisation
• graph isomorphism
• Ulam-Kelly Reconstruction Conjecture

Innovation

• Homomorphism Indistinguishability
• Axiomatic Characterisation
• Topological Structure
• Graph Parameters
• Reconstruction Conjecture