The indecomposable tournaments T with |W5(T)| = |T|-2

We consider a tournament T = (V, A). For X subset of V, the subtournament of T induced by X is T [X] = (X, A boolean AND (X x X)). An interval of T is a subset X of V such that, for a, b is an element of X and x is an element of V \ X, (a, x) is an element of A if and only if (b, x) is an element of A. The trivial intervals of T are empty set, {x} (x is an element of V) and V. A tournament is indecomposable if all its intervals are trivial. For n >= 2, W2n+1 denotes the unique indecomposable tournament defined on {0, ... , 2n} such that W2n+1[{0, ... , 2n - 1}] is the usual total order. Given an indecomposable tournament T, W-5(T) denotes the set of v is an element of V such that there is W subset of V satisfying v is an element of W and T[W] is isomorphic to W-5. Latka [6] characterized the indecomposable tournaments T such that W-5(T) = empty set. The authors [1] proved that if W-5(T) not equal empty set, then vertical bar W-5(T)vertical bar >= vertical bar V vertical bar - 2. In this note, we characterize the indecomposable tournaments T such that vertical bar W5(T)vertical bar = vertical bar V vertical bar - 2. (c) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.