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Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids

Summary: Cubic Time Recognition of Cocircuit Graphs of
Uniform Oriented Matroids
Stefan Felsner, Ricardo G´omez, Kolja Knauer,
Juan Jos´e Montellano-Ballesteros, Ricardo Strausz
July 20, 2010
We present an algorithm which takes a graph as input and decides
in cubic time if the graph is the cocircuit graph of a uniform oriented
matroid. In the affirmative case the algorithm returns the set of signed
cocircuits of the oriented matroid. This improves an algorithm proposed
by Babson, Finschi and Fukuda.
Moreover we strengthen a result of Montellano-Ballesteros and Strausz
characterizing cocircuit graphs of uniform oriented matroids in terms of
crabbed connectivity.
1 Introduction
The cocircuit graph is a natural combinatorial object associated with oriented
matroids. In the case of spherical pseudoline-arrangements, i.e., rank 3 oriented
matroids, its vertices are the intersection points of the lines and two points
share an edge if they are adjacent on a line. More generally, the Topological
Representation Theorem of Folkman and Lawrence [5] says that every oriented


Source: Aíza, Ricardo Gómez - Instituto de Matemáticas, Universidad Nacional Autónoma de México


Collections: Mathematics