{
  "id": "1407.4066",
  "version": "v2",
  "published": "2014-07-15T17:39:28.000Z",
  "updated": "2015-03-17T03:29:57.000Z",
  "title": "Minors and dimension",
  "authors": [
    "Bartosz Walczak"
  ],
  "comment": "Major revision of the exposition and the proof",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-15T17:39:28.000Z",
      "abstract": "Streib and Trotter proved in 2012 that posets with bounded height and with planar cover graphs have bounded dimension. Recently, Joret et al. proved that the dimension is bounded for posets with bounded height whose cover graphs have bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed (topological) minor have bounded dimension. This generalizes both the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx structural decomposition theorems.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-17T03:29:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05C35"
    ],
    "keywords": [
      "bounded height",
      "grohe-marx structural decomposition theorems",
      "bounded dimension",
      "cover graphs exclude",
      "planar cover graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4066W"
    }
  }
}