A Mihalisin–Klee Theorem for Fans

Locke, Rachel E.; Morris, Walter D.

doi:10.1007/s00454-016-9787-1

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2021

2022

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Cited by 2 publications

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“…In dimension 3, the Holt-Klee condition is also known to be sufficient for an acyclic orientation of the graph of a polytope with a unique source and sink on every face to be d-polytopal. (See [19], [17]).…”

Section: Introductionmentioning

confidence: 99%

On the Holt-Klee Property for Oriented Matroid Programming

Morris¹

2021

Preprint

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The Holt-Klee theorem says that the graph of a d-polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d + 1 on n + 2 elements, which include those from d-polytopes with n facets, admit d independent monotone paths from source to sink if d ≤ 4. This was previously only known to hold for d ≤ 3 and n ≤ 6.

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Section: Introductionmentioning

confidence: 99%