{
  "abstract": "We present compact SAT encodings and certifying exact solvers for boxicity and cubicity. The boxicity of a graph is the minimum dimension in which it is the intersection graph of axis-aligned boxes; cubicity is defined analogously using unit cubes. Grid-based formulations face a geometric discretization bottleneck, growing rapidly with the target dimension. We avoid this bottleneck by encoding the total order of interval endpoints in each coordinate. For an $n$-vertex graph and target dimension $d$, our boxicity encoding uses $O(n^2d)$ variables and $O(n^3d)$ clauses, both linear in $d$. Requiring the left- and right-endpoint orders to agree yields a cubicity encoding of the same asymptotic size. Satisfying models are decoded into independently checked geometric representations, while unsatisfiable instances used as lower-bound evidence carry mechanically verified DRAT certificates. In a controlled evaluation with 186 trials per encoding on standard and structured graph families, the order encoding solves 96.8% of trials, compared with 71.0% for the known grid formulation, and reduces mean PAR-2 time from 178.21 to 21.70 seconds. The pipeline also certifies exact values for several named graphs. We turn one such value into a reusable obstruction by proving that every Kneser graph $K(n,k)$ with $k\\ge3$ and $n\\ge2k+1$ contains an induced Desargues graph. The resulting lower bound, together with an explicit three-dimensional representation of $K(7,3)$, establishes $\\operatorname{boxicity}(K(7,3))=3$. This disproves the conjecture of Caoduro and Sebo that $\\operatorname{boxicity}(K(n,k))=n-k$ for all $k\\ge3$ and $n\\ge2k+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22241",
  "author_total": 4,
  "authors": [
    {
      "name": "L. Sunil Chandran"
    },
    {
      "name": "Rishikesh Gajjala"
    },
    {
      "name": "Suraj Kumar Sahoo"
    },
    {
      "name": "Aatman Supkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7y1dkgmy2qnx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22241",
  "fulltext_url": null,
  "id": "7y1dkgmy2qnx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22241",
  "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.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222759545",
  "pdf_url": "https://arxiv.org/pdf/2610.22241",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7y1dkgmy2qnx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22241",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22241"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Geometric Graph Representations via Compact SAT Encodings",
  "title_text": "Geometric Graph Representations via Compact SAT Encodings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/7y1dkgmy2qnx3",
  "venue": "arXiv",
  "versions": []
}