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 the four known necessary conditions for a polytopal digraph to be an LP‑digraph (the others being acyclicity, the unique‑sink orientation (USO), and the Holt‑Klee property). While the first three conditions have been shown to be relatively weak, the shelling property is more subtle and its exact role in characterising LP‑digraphs remains unclear.

The authors begin by revisiting the original definition of the shelling property introduced by Avis and Moriyama (2009). The classic formulation requires the existence of a vertex ordering that induces a shelling on every face of the polytope. The paper proposes a seemingly stronger version: every possible vertex ordering must induce a shelling on each face. Through a careful combinatorial argument they prove that the two formulations are in fact equivalent. This equivalence is crucial because it allows the authors to work with the stronger, more convenient formulation in the subsequent constructions.

The central contribution is a general construction that produces infinite families of polytopal digraphs which satisfy acyclicity, USO, and the Holt‑Klee property but fail the shelling property. The construction starts from the minimal counterexample known in dimension four with six vertices (the smallest 4‑polytope whose unique sink is simple and whose digraph violates shelling). The key insight is that if the unique sink of a 4‑polytope is simple—i.e., incident to exactly four edges—then one can “inflate’’ the polytope to higher dimensions while preserving the three good properties and deliberately preserving the shelling failure.

The inflation operation proceeds as follows. Given a 4‑polytope (P) with a simple sink, one adds a new dimension and a new vertex adjacent to the sink in a way that the orientation of all existing edges is left unchanged. The new vertex is then connected to a carefully chosen set of existing vertices so that any linear extension of the vertex order still contains a face whose induced sub‑order is not a shelling. Repeating this operation (k) times yields a (d = 4 + k) dimensional polytope with (n = n_0 + k) vertices (where (n_0) is the original vertex count) that still lacks a shelling. Consequently, for any dimension (d \ge 4) and any number of vertices (n \ge n_0 + d - 4) there exists a polytopal digraph with the three good properties but without a shelling. This dramatically extends the earlier result of Avis, Miyata and Moriyama (2009), which required (n \ge d + 2).

The paper also examines the strength of the shelling condition for the family of (d)-crosspolytopes. Develin (2004) gave a complete characterisation of LP‑orientations of crosspolytopes, showing that every LP‑orientation satisfies acyclicity, USO and the Holt‑Klee property. The authors prove that, for crosspolytopes, the shelling property is automatically satisfied by any orientation that meets the first three conditions. Hence, in this highly symmetric class the shelling condition is not only necessary but also sufficient for an LP‑orientation. This contrast highlights that the shelling property can be either a weak or a strong constraint depending on the underlying polytope’s combinatorial structure.

In summary, the paper makes three major advances: (1) it clarifies the definition of the shelling property and shows the equivalence of two natural formulations; (2) it provides a versatile construction that, starting from any 4‑polytope with a simple sink, yields arbitrarily high‑dimensional polytopes whose digraphs satisfy all known necessary LP‑conditions except shelling; and (3) it demonstrates that for crosspolytopes the shelling condition coincides with the LP‑orientation characterisation, thereby underscoring its variable strength across polytope families. These results deepen our understanding of why a complete characterisation of LP‑digraphs remains elusive and point toward new avenues for investigating the interplay between combinatorial geometry and linear optimisation.