{
  "abstract": "The Weisfeiler-Leman (WL) algorithm tests graph isomorphism by assigning colours to $d$-tuples of vertices and iteratively refining the colouring according to neighbourhood configurations until it stabilizes. Increasing $d$ allows the test to detect finer structural differences between graphs, but the computational cost grows exponentially with $d$. The central question is which dimension allows us to distinguish nonisomorphic graphs. We relate a sufficient such dimension to the structure of a graph $G$ through a rank decomposition over $\\mathbb F_2$. The two parameters governing the bound are the maximum cut rank $w$ and the fork load $\\ell$, which is the largest sum of the parent and two child cut ranks at a fork in the decomposition tree. We prove that WL in dimension $\\max\\{\\ell+1,2w+2\\}$ distinguishes every finite nonempty vertex-coloured graph $G$ with such a decomposition from every graph not isomorphic to $G$. Consequently, for $k\\geq1$, graphs of rank width at most $k$ can be distinguished from every nonisomorphic graph in dimension $3k+1$, graphs of linear rank width at most $k$ in dimension $2k+2$, and edgeless graphs in dimension one. Uncoloured Cai-Fürer-Immerman (CFI) constructions give linear lower bounds for both width parameters, so the upper bounds are tight up to constant factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10800",
  "author_total": 1,
  "authors": [
    {
      "name": "Antonios Kalampakas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3px33x4swqkyw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10800",
  "fulltext_url": null,
  "id": "3px33x4swqkyw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10800",
  "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",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221119670",
  "pdf_url": "https://arxiv.org/pdf/2610.10800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3px33x4swqkyw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Decomposition Profiles and Weisfeiler-Leman Dimension for Graphs of Bounded Rank Width",
  "title_text": "Decomposition Profiles and Weisfeiler-Leman Dimension for Graphs of Bounded Rank Width",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/3px33x4swqkyw",
  "venue": "arXiv",
  "versions": []
}