{
  "abstract": "In this paper we study pairwise compatibility graphs whose witness tree is required to be an unweighted caterpillar. This is a natural restriction of PCGs, in the same line of research as the study of star witnesses, which led to the class of double-threshold graphs. In the caterpillar case, we show that this restriction is equivalent to a one-dimensional difference-labelling problem: a graph admits such a representation if and only if its vertices can be assigned integer labels so that adjacency is determined exactly by whether the absolute difference of two labels belongs to a given interval. Using this characterization, we derive structural properties of the class, including bounded twin-width, and study its relation with familiar graph families. We show that the class contains cycles and two-dimensional grids, while it excludes natural obstructions such as the subdivided star $S_{2,2,2,2,2}$, the cube graph $Q_3$, the wheel $W_6$, and the complement of the $3$-sunlet. We also consider restricted representations, including centipede witnesses and representations with a given interval and we give a polynomial-time recognition algorithm when the interval is fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14681",
  "author_total": 4,
  "authors": [
    {
      "name": "Cyriac Antony"
    },
    {
      "name": "Manuel Lafond"
    },
    {
      "name": "Angelo Monti"
    },
    {
      "name": "Blerina Sinaimeri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/esj52e33b5c88.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14681",
  "fulltext_url": null,
  "id": "esj52e33b5c88",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14681",
  "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|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221770150",
  "pdf_url": "https://arxiv.org/pdf/2610.14681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/esj52e33b5c88.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14681"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pairwise Compatibility Graphs of Unweighted Caterpillars",
  "title_text": "Pairwise Compatibility Graphs of Unweighted Caterpillars",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/esj52e33b5c88",
  "venue": "arXiv",
  "versions": []
}