{
  "abstract": "The Weisfeiler-Leman algorithms WL(k) for $k\\ge1$ form a hierarchy of increasingly powerful tests for graph isomorphism. We study how the equivalence relations they induce on graphs transfer through a graph operation $Φ$: for a given $k$, what level of WL-equivalence between graphs $H_1$ and $H_2$ guarantees WL(k)-equivalence of graphs $Φ(H_1)$ and $Φ(H_2)$? We call the least such level the transfer parameter $τ_Φ(k)$ and determine it exactly for the line graph operation $Φ= L$: $τ_L(k)=2k+1$ for $k\\ge2$, while $τ_L(1)=1$. We also extend these results to the $(r,S)$-line graphs, which generalize the line graph from edges to $r$-cliques by connecting two cliques when their intersection size lies in a prescribed set $S$. For these operations we prove that $τ_{r,S}(1)=r-1$ for every $S$ and, for certain choices of $S$, that $τ_{r,S}(k)=rk+r-1$ for $k\\ge r$, and we bound the transfer parameter in the remaining range. Our upper bounds on transfer parameters follow from a treewidth analysis of diagrammatic expansions of homomorphism counts into $Φ(H)$ as linear combinations of homomorphism counts into $H$. The matching lower bounds combine explicit constructions of witness graphs with the homomorphism-distinguishing closedness of graphs of bounded treewidth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05366",
  "author_total": 2,
  "authors": [
    {
      "name": "Dmitriy Kunisky"
    },
    {
      "name": "Hamilton Sawczuk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fem7b7ezxpg4q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05366",
  "fulltext_url": null,
  "id": "fem7b7ezxpg4q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05366",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fem7b7ezxpg4q.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Transfer of Weisfeiler-Leman equivalence under graph operations",
  "title_text": "Transfer of Weisfeiler-Leman equivalence under graph operations",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/fem7b7ezxpg4q",
  "venue": "arXiv",
  "versions": []
}