*-graphs of Vertices of the Generalized Transitive Tournament Polytope

A nonnegative matrix T = (tij)~,j= 1 is a generalized transitive tournament matrix (GTT matrix) if tii = 0, t o = 1 tj~ for i # j , and 1 ~< t o + tjk + t~ ~< 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 E {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.