Generation of Oriented Matroids—A Graph Theoretical Approach

Abstract

We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for non-uniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore, we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs finally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

Matroid Enumeration for Incidence Geometry

“…Therefore the number of oriented matroids is much larger than that of matroids. Oriented matroids of rank r on n elements are determined for r = 3, n ≤ 10 and r = 4, n ≤ 8 by Finschi and Fukuda [18][19][20]. In the case of uniform oriented matroids, those of rank 3 on n ≤ 11 elements were determined by Aichholzer and Krasser [2].…”

“…Our matroid enumeration algorithm falls into this class of algorithm. For enumeration of oriented matroids (a signed extension of matroids) Finschi and Fukuda [18][19][20] succeeded in this approach. This encourages us to apply their idea to enumeration of matroids.…”

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

“…There are quite a number of software packages to generate oriented matroids (for instance [5,8,13,16]). These packages use one of two different approaches: The programs by Bokowski and Guedes de Oliveira and by Finschi construct oriented matroids by using single element extensions, whereas the other programs try to construct the oriented matroids globally by filling in the chirotopes.…”

On the Number of Simple Arrangements of Five Double Pseudolines

“…However it is possible, as explained in [14], to reduce this axiomatization to a (relatively) short list of axioms with a clear combinatorial or geometrical interpretation of each axiom in a maner very similar to the known simple axiomatizations of pseudoline arrangements [3,15]. This open the door to the generation of double pseudoline arrangements with prescribed properties using, for example, satisfiability solvers as proposed in [16,17], see also [5,7,2,8,6]. In particular it will be interesting to generate the arrangements that maximize the number of connected components of the intersection of the Möbius strips surrounded by the double pseudolines.…”