{
  "abstract": "Let $m,n \\geq 3$. A set of vertices $S$ of the rectangular grid graph $P_m \\square P_n$ is resolving if the taxicab distance vectors of the vertices of $P_m \\square P_n$ with respect to $S$ are pairwise distinct. A resolving set $S$ is minimal if no proper subset of $S$ is resolving, and a minimal resolving set of cardinality $k$ is called a $k$-minimal. We extend the work of Andersen et al., who characterized $3$-minimals, established the maximum cardinality $2\\min(m,n)-2$, and posed the complete characterization and enumeration of minimal resolving sets for grids as an open problem, and Adar and Epstein, who showed that $3$ is the only possible odd cardinality and that every minimal resolving set of cardinality at least $4$ can be ordered to form a sequence corresponding to a zigzag sequence. We provide a recursive construction that generates exactly the minimal resolving sets of cardinality at least $4$ for grids. We derive from the construction closed-form formulas to enumerate the $k$-minimals for all even $4 \\leq k \\leq 2\\min(m,n) -2$. Together with the known characterizations for cardinalities $2$ and $3$ and a direct enumeration of the $3$-minimals, this yields a complete characterization and enumeration of the minimal resolving sets of rectangular grid graphs of every possible cardinality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35896",
  "author_total": 1,
  "authors": [
    {
      "name": "Caroline Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q7c8qvyn0nd7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35896",
  "fulltext_url": null,
  "id": "0q7c8qvyn0nd7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35896",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7214963583",
  "pdf_url": "https://arxiv.org/pdf/2609.35896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q7c8qvyn0nd7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.35896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Minimal Resolving Sets in Rectangular Grid Graphs: A Complete Characterization and Enumeration",
  "title_text": "Minimal Resolving Sets in Rectangular Grid Graphs: A Complete Characterization and Enumeration",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0q7c8qvyn0nd7",
  "venue": "arXiv",
  "versions": []
}