Publication

The Holt-Klee condition for oriented matroids

2009
Journal Articles

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. (C) 2008 Elsevier Ltd. All rights reserved.

Official source

https://infoscience.epfl.ch/entities/publication/c705d918-8ffc-49c3-bbc3-a0b6ffe90c9f

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2009
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/c705d918-8ffc-49c3-bbc3-a0b6ffe90c9f

About this result

Related concepts (32)

Matroid

In combinatorics, a branch of mathematics, a matroid ˈmeɪtrɔɪd is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.

Oriented matroid

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered. All oriented matroids have an underlying matroid.

Graphic matroid

In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.

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