{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07927",
  "author_total": 6,
  "authors": [
    {
      "name": "Josse van Dobben de Bruyn"
    },
    {
      "name": "Jérémie Marquès"
    },
    {
      "name": "David E. Roberson"
    },
    {
      "name": "Tim Seppelt"
    },
    {
      "name": "Gian Luca Spitzer"
    },
    {
      "name": "Peter Zeman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77v47t4z4k06y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07927",
  "fulltext_url": null,
  "id": "77v47t4z4k06y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07927",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221030908",
  "pdf_url": "https://arxiv.org/pdf/2610.07927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77v47t4z4k06y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Topological and Geometric Perspectives on Homomorphism Indistinguishability",
  "title_text": "Topological and Geometric Perspectives on Homomorphism Indistinguishability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/77v47t4z4k06y",
  "venue": "arXiv",
  "versions": []
}