*-graphs of vertices of the generalized transitive tournament polytope

A nonnegative matrix T = (t(ij))(i,j=1)(n), is a generalized transitive tournament matrix (GTT matrix) if t(ii) = 0, t(ij) = 1-t(ji) for i not equal j, and 1 less than or equal to t(ij) + t(jk) + t(ki) less than or equal to 2 for i, j, k pairwise distinct. The problem we are interested in is the characterization of the set of vertices of the polytope {GTT}(n), of all GTT matrices of order n. In 1992, Brualdi and Hwang introduced the *-graph associated to each T is an element of {GTT}(n),. We characterize the comparability graphs of n vertices which are the *-graphs of some vertex of {GTT}(n),. As an application of the theoretical work we conclude that no comparability graph of at most 6 vertices and with at least one edge is the *-graph of a vertex. In order to obtain the set of all vertices of {GTT}(6) it only remains to analyse two noncomparability graphs.