The Holt-Klee condition for oriented matroids

Komei Fukuda, Sonoko Moriyama, Yoshio Okamoto

研究成果: Article査読

6 被引用数 (Scopus)

抄録

Holt and Klee have recently shown that every (generic) LP orientation of the graph of a d-polytope satisfies a directed version of the d-connectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any non-HK OM. This leads to a fundamental question as to whether the Holt-Klee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8-element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no non-HK OM in this class.

本文言語English
ページ(範囲)1854-1867
ページ数14
ジャーナルEuropean Journal of Combinatorics
30
8
DOI
出版ステータスPublished - 2009 11

ASJC Scopus subject areas

  • 離散数学と組合せ数学

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