{
  "id": "2405.06755",
  "version": "v1",
  "published": "2024-05-10T18:16:38.000Z",
  "updated": "2024-05-10T18:16:38.000Z",
  "title": "Counterexamples regarding linked and lean tree-decompositions of infinite graphs",
  "authors": [
    "Sandra Albrechtsen",
    "Raphael W. Jacobs",
    "Paul Knappe",
    "Max Pitz"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Kriz and Thomas showed that every (finite or infinite) graph of tree-width $k \\in \\mathbb{N}$ admits a lean tree-decomposition of width $k$. We discuss a number of counterexamples demonstrating the limits of possible generalisations of their result to arbitrary infinite tree-width. In particular, we construct a locally finite, planar, connected graph that has no lean tree-decomposition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-10T18:16:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C63",
      "05C05",
      "05C83",
      "05C40"
    ],
    "keywords": [
      "lean tree-decomposition",
      "infinite graphs",
      "counterexamples regarding",
      "arbitrary infinite tree-width",
      "connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}