Families 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. Avis, Miyata and Moriyama(2009) constructed for each $d \ge 4$ and $n \ge d+2$, a $d$-polytope $P$ with $n$ vertices which has a polytopal digraph which is an acyclic USO that satisfies the Holt-Klee property, but does not satisfy the shelling property. The construction was based on a minimal such example, which has $d=4$ and $n=6$. In this paper we explore the shelling condition further. First we give an apparently stronger definition of the shelling property, which we then prove is equivalent to the original definition. Using this stronger condition we are able to give a more general construction of such families. In particular, we show that given any 4-dimensional polytope $P$ with $n_0$ vertices whose unique sink is simple, we can extend $P$ for any $d \ge 4$ and $n \ge n_0 + d-4$ to a $d$-polytope with these properties that has $n$ vertices. Finally we investigate the strength of the shelling condition for $d$-crosspolytopes, for which Develin (2004) has given a complete characterization of LP orientations.

💡 Research Summary

The paper investigates the “shelling property,” one of four necessary conditions for a polytopal digraph G(P) to be an LP‑digraph (the others being acyclicity, unique‑sink orientation (USO), and the Holt‑Klee property). While the first three conditions have been well studied, the shelling property—originating from polytope theory—remains less understood, especially in dimensions d ≥ 4.

First, the authors revisit the original definition of the shelling property (Avis–Moriyama 2009), which requires the existence of a topological ordering r of the vertices such that the corresponding ordering r* of the facets of the combinatorial polar P* forms a shelling. They propose a seemingly stronger definition: “for any topological sort r of G(P), the induced facet order is a shelling of P*.” Using Lemma 6 (which characterizes the intersection of a facet with previously ordered facets in terms of incoming edges) and Theorem 7, they prove that the two definitions are equivalent. This equivalence simplifies verification: it suffices to check the property for a single topological sort.

Next, the paper presents a concrete 4‑dimensional example Ω. Ω has eight vertices and ten facets; directing every edge from the smaller‑indexed vertex to the larger yields an acyclic USO that also satisfies the Holt‑Klee condition (there are three vertex‑disjoint source‑to‑sink paths in each 2‑dimensional face and four such paths across the whole polytope). However, the unique topological order (1,2,…,10) does not give a shelling of the polar Ω*. Specifically, the intersection of the third facet with the union of the first two facets consists of two 2‑faces meeting only at a single vertex, violating the “beginning segment” requirement of a shelling. Hence Ω provides a new minimal X‑type graph (an acyclic USO with Holt‑Klee but without the shelling property).

The central contribution is a general construction that extends any 4‑polytope P₀ possessing an X‑type graph whose unique sink is a simple vertex to higher dimensions and larger vertex counts. Theorem 4 states: for any d ≥ 4 and any n ≥ n₀ + d − 4, there exists a d‑polytope P with n vertices that also has an X‑type graph. The construction proceeds by repeatedly applying two operations:

1. Pyramid operation – adds a new apex vertex, raising the dimension by one while preserving acyclicity, USO, and Holt‑Klee.
2. Stacking (truncation) operation – inserts additional vertices on existing facets to increase the vertex count without destroying the three properties.

Because the shelling property is not restored by these operations, the resulting family yields infinitely many examples across all admissible (d, n) pairs.

Finally, the authors turn to d‑crosspolytopes. Develin (2004) introduced “pair sequences” that completely characterize LP‑orientations of crosspolytopes. Using Develin’s results, the paper shows that for crosspolytopes the shelling property is equivalent to being an LP‑orientation; consequently, any acyclic USO that fails the shelling condition cannot be an LP‑digraph on a crosspolytope. This demonstrates that the shelling property can be strictly stronger than the other three conditions in certain families, providing a quantitative measure of its discriminating power.

In summary, the paper (1) clarifies the definition of the shelling property and proves its equivalence to a stronger formulation, (2) supplies a new minimal 4‑dimensional X‑type example, (3) furnishes a scalable construction that generates X‑type graphs for arbitrary dimensions d ≥ 4 and vertex counts n ≥ d + 2, and (4) confirms that for crosspolytopes the shelling property exactly captures LP‑orientability. These results deepen our structural understanding of polytopal digraphs and highlight the shelling property as a pivotal obstacle in characterizing LP‑digraphs in higher dimensions.