{
  "id": "1204.2513",
  "version": "v1",
  "published": "2012-04-11T18:26:56.000Z",
  "updated": "2012-04-11T18:26:56.000Z",
  "title": "The {-3}-reconstruction and the {-3}-self duality of tournaments",
  "authors": [
    "Mouna Achour",
    "Youssef Boudabbous",
    "Abderrahim Boussairi"
  ],
  "comment": "To appear in Ars Combinatoria; Published in Ars Combinatoria on April 2010",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let T = (V,A) be a (finite) tournament and k be a non negative integer. For every subset X of V is associated the subtournament T[X] = (X,A\\cap (X \\timesX)) of T, induced by X. The dual tournament of T, denoted by T\\ast, is the tournament obtained from T by reversing all its arcs. The tournament T is self dual if it is isomorphic to its dual. T is {-k}-self dual if for each set X of k vertices, T[V \\ X] is self dual. T is strongly self dual if each of its induced subtournaments is self dual. A subset I of V is an interval of T if for a, b \\in I and for x \\in V \\ I, (a,x) \\in A if and only if (b,x) \\in A. For instance, \\varnothing, V and {x}, where x \\in V, are intervals of T called trivial intervals. T is indecomposable if all its intervals are trivial; otherwise, it is decomposable. A tournament T', on the set V, is {-k}-hypomorphic to T if for each set X on k vertices, T[V \\ X] and T'[V \\ X] are isomorphic. The tournament T is {-k}-reconstructible if each tournament {-k}-hypomorphic to T is isomorphic to it. Suppose that T is decomposable and | V |\\geq 9. In this paper, we begin by proving the equivalence between the {-3}-self duality and the strong self duality of T. Then we characterize each tournament {-3}-hypomorphic to T. As a consequence of this characterization, we prove that if there is no interval X of T such that T[X] is indecomposable and | V \\ X |\\leq 2, then T is {-3}-reconstructible. Finally, we conclude by reducing the {-3}-reconstruction problem to the indecomposable case (between a tournament and its dual). In particular, we find and improve, in a less complicated way, the results of [6] found by Y. Boudabbous and A. Boussairi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-11T18:26:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isomorphic",
      "strong self duality",
      "dual tournament",
      "strongly self dual",
      "trivial intervals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.2513A"
    }
  }
}