ON THE (-K)-HALF RECONSTRUCTIBILITY OF FI NITE TOURNAMENTS

Given a tournament T, we define the dual T of T by T* (x,y) = T(y,x). A tournament T' is hemimorphic to T if it is isomorphic to T or T. A tournament defined on n elements is (-k)-reconstructible (resp. (-k)- half-reconstructible) if it is determined up to isomorphism (resp. up to hemimorphism), by its restrictions to subsets of (n - k) elements. From [2] follows the (-k)-half-reconstructibility of finite tournament s (with n greater than or equal to (7 + k) elements), for all k greate r than or equal to 7. In this Note, we establish the (-k)-half-reconst urctibility of finite tournaments (with n > (12 + k) elements), for al l k is an element of {4,5,6}. We then connect the problems of the (-3) - and the (-2)-half-reconstruction of these tournaments to two problem s (yet open) of reconstruction. Finally, by using counterexamples of P .K. Stockmeyer [14], we show that, generally, the finite tournaments a re not (-k)-half-reconstructible. (C) Academie des Sciences/Elsevier, Paris.