{
  "id": "1406.3397",
  "version": "v2",
  "published": "2014-06-13T01:03:56.000Z",
  "updated": "2014-10-03T10:00:07.000Z",
  "title": "On the dimension of posets with cover graphs of treewidth $2$",
  "authors": [
    "Gwenaël Joret",
    "Piotr Micek",
    "William T. Trotter",
    "Ruidong Wang",
    "Veit Wiechert"
  ],
  "comment": "v2: minor changes",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Bir\\'o, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth $2$. We show that it is indeed the case: Every such poset has dimension at most $2554$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-13T01:03:56.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-03T10:00:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cover graph",
      "arbitrarily large dimension",
      "boundary case",
      "well-known construction",
      "outerplanar"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3397J"
    }
  }
}