On the Cocircuit Graph of an Oriented Matroid

In this paper we consider the cocircuit graph G M 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 G M. However, we show that if the vertex set resp. edge set] of G M is properly labeled by the hyperplanes resp. colines] of M, G M determines W. Also we prove that, when M is uniform, the cocicuit 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 one skeleton of a simple convex polytope determines its face lattice.