{
  "abstract": "For a planar graph $G$ with at least one edge, let $ρ(G)$ be the infimum of the ratio of the longest to the shortest edge length over all planar straight-line drawings of $G$. We prove that every $n$-vertex $2$-tree $G$ satisfies $ρ(G)\\le4+5\\lfloor\\log_2(2n-3)\\rfloor$. This improves the known $O(\\sqrt n)$ upper bound and, together with the lower bound of Blažej, Fiala and Liotta, shows that the maximum of $ρ(G)$ over $n$-vertex $2$-trees is $Θ(\\log n)$. The proof adapts Biedl's recursive bar construction using nonintegral heights and applies a height-preserving straightening theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28030",
  "author_total": 3,
  "authors": [
    {
      "name": "Zahra Sabaei"
    },
    {
      "name": "Fatemeh Emami"
    },
    {
      "name": "Yasmin Tousinejad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/68vxjj7j1dckd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28030",
  "fulltext_url": null,
  "id": "68vxjj7j1dckd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28030",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7223142186",
  "pdf_url": "https://arxiv.org/pdf/2610.28030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/68vxjj7j1dckd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.28030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Logarithmic edge-length ratios for planar drawings of 2-trees",
  "title_text": "Logarithmic edge-length ratios for planar drawings of 2-trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/68vxjj7j1dckd",
  "venue": "arXiv",
  "versions": []
}