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  Geometry © 2000 Springer-Verlag New York Inc. On the Cocircuit Graph of an Oriented Matroid ∗

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by R. Cordovil, K. Fukuda, A. Guedes De Oliveira
http://www.math.ist.utl.pt/~rcordov/CGOM.pdf
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Abstract:

Abstract. In this paper we consider the cocircuit graph GM of an oriented matroid M, the 1-skeleton of the cell complex W formed by the span of the cocircuits of M. In general, W is not determined by GM. However, we show that if the vertex set (resp. edge set) of GM is properly labeled by the hyperplanes (resp. colines) of M,GM determines W. Also we prove that, when M is uniform, the cocircuit graph together with all antipodal pairs of vertices being marked determines W. These results can be considered as variations of Blind–Mani’s theorem that says the 1-skeleton of a simple convex polytope determines its face lattice.

Citations

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1 and in revised form March 2 – August - 1998
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