A nonnegative matrix T = ( t ij ) i, j= t n is a generalized transitive tournament matrix (GTT matrix) if t jj = 0, t ij = 1 − t ji for i ≠ j, and 1 ⩽ t ij + t jk + t ki ⩽ 2 for i, j, k pairwise distinct. An approach to the problem of characterize the set of vertices of the polytope GTT n of all GTT matrices of order n was the introduction by Brualdi and Hwang of the ∗-graph associated to each T ϵ GTT n . We introduce a new graph which generalize the ∗-graph. The new graph will be employed to develop a computable criterion for determine whether any given GTT matrix of order n is or not a vertex of GTT n . A consequence of the criterion is that if T is a vertex of GTT n . with small number, r, of different entries then we have strong restrictions for the possible entries of T. Namely, if r ⩽ 6 then the set of entries of T is equal to 0, 1, 0, 1, 0, 1 2 ,1,0, 1 3 , 2 3 ,1, 0, 1 3 , 1 2 , 2 3 ,1, 0, 1 4 , 1 2 , 3 4 ,1, 0, 1 6 , 1 3 , 2 3 , 5 6 ,1, or 1 5 , 2 5 , 3 5 , 4 5 ,1.