{
  "id": "1709.00313",
  "version": "v1",
  "published": "2017-09-01T13:53:49.000Z",
  "updated": "2017-09-01T13:53:49.000Z",
  "title": "A Simple Proof Characterizing Interval Orders with Interval Lengths between 1 and $k$",
  "authors": [
    "Simona Boyadzhiyska",
    "Garth Isaak",
    "Ann Trenk"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A poset $P= (X, \\prec)$ has an interval representation if each $x \\in X$ can be assigned a real interval $I_x$ so that $x \\prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \\emph{interval orders}. Fishburn proved that for any positive integer $k$, an interval order has a representation in which all interval lengths are between $1$ and $k$ if and only if the order does not contain $\\mathbf{(k+2)+1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-01T13:53:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A99",
      "05C62"
    ],
    "keywords": [
      "simple proof characterizing interval orders",
      "interval lengths",
      "digraph model",
      "real interval",
      "interval representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}