{
  "id": "1105.1611",
  "version": "v5",
  "published": "2011-05-09T09:59:21.000Z",
  "updated": "2014-09-01T10:28:54.000Z",
  "title": "Connectivity and tree structure in finite graphs",
  "authors": [
    "Johannes Carmesin",
    "Reinhard Diestel",
    "Fabian Hundertmark",
    "Maya Stein"
  ],
  "comment": "31 pages",
  "journal": "Combinatorica 34 (2014), pp 11-46",
  "doi": "10.1007/s00493-014-2898-5",
  "categories": [
    "math.CO"
  ],
  "abstract": "Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the $k$-blocks -- the maximal vertex sets that cannot be separated by at most $k$ vertices -- of a graph $G$ live in distinct parts of a suitable tree-decomposition of $G$ of adhesion at most $k$, whose decomposition tree is invariant under the automorphisms of $G$. This extends recent work of Dunwoody and Kr\\\"on and, like theirs, generalizes a similar theorem of Tutte for $k=2$. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all $k$ simultaneously, all the $k$-blocks of a finite graph.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-02-04T09:50:57.000Z",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-09-01T10:28:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C05",
      "05C83",
      "G.2.2"
    ],
    "keywords": [
      "finite graph",
      "tree structure",
      "connectivity",
      "maximal vertex sets",
      "separation system contains"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.1611C"
    }
  }
}