154 research outputs found
On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid
The first author introduced the circuit-cocircuit reversal system of an
oriented matroid, and showed that when the underlying matroid is regular, the
cardinalities of such system and its variations are equal to special
evaluations of the Tutte polynomial (e.g., the total number of
circuit-cocircuit reversal classes equals , the number of bases of
the matroid). By relating these classes to activity classes studied by the
first author and Las Vergnas, we give an alternative proof of the above results
and a proof of the converse statements that these equalities fail whenever the
underlying matroid is not regular. Hence we extend the above results to an
equivalence of matroidal properties, thereby giving a new characterization of
regular matroids.Comment: 7 pages. v2: simplified proof, with new statements concerning other
special evaluations of the Tutte polynomia
Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids
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 about
crabbed connectivity of cocircuit graphs of uniform oriented matroids.Comment: 9 page
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