Authors’ abstract: A nonnegative matrix \(T=(t_{ij})^n_{i,j=1}\) is a generalized transitive tournament matrix (GTT matrix) if \(t_{ii}=0\), \(t_{ij}= 1-t_{ji}\) for \(i\neq j\), and \(1\leq t_{ij} +t_{jk}+ t_{ki}\leq 2\) for \(i,j,k\) pairwise distinct. The problem we are interested in is the characterization of the set of vertices of the polytope \(\{\text{GTT}\}_n\) of all GTT matrices of order \(n\). In 1992, Brualdi and Hwang introduced the *-graph associated to each \(T \in \{\text{GTT}\}_n\), see R. A. Brualdi and G.-S. Hwang [Linear Algebra Appl. 172, 151-168 (1992; Zbl 0755.15011)]. We characterize the comparability graphs of \(n\) vertices which are the *-graphs of some vertex of \(\{\text{GTT}\}_n\). 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 \(\{\text{GTT}\}_6\) it only remains to analyse two noncomparability graphs.