{
  "abstract": "An identifying code of a graph is a dominating set $C$ such that the sets $N[v]\\cap C$ are pairwise distinct. The minimum density of an identifying code has been studied extensively for several infinite grids. It is known to be $1/4$ for the triangular grid and $7/20$ for the square grid, whereas for the hexagonal grid the problem is still open, between $5/12$ and $53/126$. These three grids are the regular members of the family of the eleven Archimedean grids. In this paper, we start a systematic study of identifying codes in this family. We determine the minimum density for five of the nine Archimedean grids for which it was not known: it is $2/3$ for the truncated hexagonal grid, $3/7$ for the truncated square grid, and $1/3$ for the trihexagonal (kagome) grid, the elongated triangular grid and the snub hexagonal grid. All lower bounds are proved by the discharging method. For the three remaining non-regular Archimedean grids, we also obtain some bounds: the minimum density lies between $3/7$ and $13/30$ for the truncated trihexagonal grid, between $1/3$ and $11/30$ for the rhombitrihexagonal grid, and between $2/7$ and $9/28$ for the snub square grid. As a by-product, we prove that every identifying code of a cubic graph with the slow growth property in which every vertex lies on exactly one triangle (resp. $4$-cycle) has density at least $2/3$ (resp. $3/7$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32192",
  "author_total": 1,
  "authors": [
    {
      "name": "Rudini Sampaio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2wgxdtpant9fx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32192",
  "fulltext_url": null,
  "id": "2wgxdtpant9fx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32192",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7223679355",
  "pdf_url": "https://arxiv.org/pdf/2610.32192",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2wgxdtpant9fx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.32192",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32192"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Identifying codes of Archimedean grids",
  "title_text": "Identifying codes of Archimedean grids",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2wgxdtpant9fx",
  "venue": "arXiv",
  "versions": []
}