Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int