A family of polytopal digraphs that do not satisfy the shelling property

A polytopal digraph G(P ) is an orientation of the skeleton of a convex polytope P . The possible non-degenerate pivot operations of the simplex method in solving a linear program over P can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known: acyclicity, unique sink orientation(USO), the Holt-Klee property and the shelling property. The shelling property was introduced by Avis and Moriyama (2009), where two examples are given in d = 4 dimensions of polytopal digraphs satisfying the first three properties but not the shelling property. The smaller of these examples has n = 7 vertices. In this paper for each d ≥ 4 and n ≥ d+ 2, we construct a polytopal digraph for a polytope P in dimension d with n vertices which is an acyclic USO that satisfies the Holt-Klee property, but does not satisfy the shelling property. It is known that such examples cannot exist for other values of n and d.