The Holt-Klee condition for oriented matroids

5Citations
Citations of this article
6Readers
Mendeley users who have this article in their library.

Abstract

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. © 2008 Elsevier Ltd. All rights reserved.

Cite

CITATION STYLE

APA

Fukuda, K., Moriyama, S., & Okamoto, Y. (2009). The Holt-Klee condition for oriented matroids. European Journal of Combinatorics, 30(8), 1854–1867. https://doi.org/10.1016/j.ejc.2008.12.012

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free