  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.
