{
  "abstract": "A $k$-bend path is a non-self-intersecting polyline that lies on a grid and consists of at most $k+1$ axis-parallel line segments. A $B_k$-CPG graph is a graph whose vertices can be represented by pairwise interiorly disjoint $k$-bend paths on a grid such that two vertices are adjacent if and only if the corresponding grid paths touch at a grid point. We prove that recognizing planar $B_0$-CPG graphs of maximum degree 8 is NP-complete, and that recognizing planar $B_1$-CPG graphs of maximum degree 11 is NP-complete. These results settle two of the three planar recognition problems left open by Champseix, Galby, Munaro, and Ries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05838",
  "author_total": 3,
  "authors": [
    {
      "name": "Bin Sun"
    },
    {
      "name": "Shou-Jun Xu"
    },
    {
      "name": "Yu Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4nnbp9vf8w5nj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05838",
  "fulltext_url": null,
  "id": "4nnbp9vf8w5nj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05838",
  "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|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7220709215",
  "pdf_url": "https://arxiv.org/pdf/2610.05838",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4nnbp9vf8w5nj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05838",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05838"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resolving Two Open Problems of Planar $B_k$-CPG Recognition: $k=0,1$",
  "title_text": "Resolving Two Open Problems of Planar B_k-CPG Recognition: k=0,1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4nnbp9vf8w5nj",
  "venue": "arXiv",
  "versions": []
}