On non-{0, 1, 1/2} extreme points of the generalized transitive tournament polytope

The acyclic tournaments of order n form the linear ordering polytope Pⁿ_LO. The generalized transitive tournaments of order n form the polytope Pⁿ_C, which contains the linear ordering polytope. It is known that the integral extreme points of Pⁿ_C coincide with those of Pⁿ_LO. Dridi showed that Pⁿ_LO = Pⁿ_LO for n ≤ 5, while for n > 5 Pⁿ_LO ⊂ Pⁿ_C. Borobia gave a complete characterization of the extreme points of Pⁿ_C with values in {0, I, 1/2}. It was mentioned by Brualdi and Hwang that no extreme points of Pⁿ_C with values not in {0, 1, 1/2) are known. In this paper we present a method for obtaining a family of extreme points of Pⁿ_C with values not in {0, 1, 1/2}. We also prove that these non-half-integral extreme points of Pⁿ_C violate certain diagonal inequalities which are facet defining for Pⁿ_LO.