On the cocircuit graph of an oriented matroid
The authors consider the cocircuit graph \(G_{\mathcal M}\) of an oriented matroid \(\mathcal M\), the 1-skeleton of the cell complex \(\mathcal W\) formed by the span of the cocircuits of \(\mathcal M\). In general, \(\mathcal W\) is not determined by \(G_{\mathcal M}\). The main result of the paper is the proof that if the vertex set (resp. edge set) of \(G_{\mathcal M}\) is properly labeled by the hyperplanes (resp. colines) of \(\mathcal M\), \(G_{\mathcal W}\) determines \(\mathcal W\). The authors also prove that, when \(\mathcal M\) is uniform, the cocircuit graph together with all antipodal pairs of vertices being marked determines \(\mathcal W\). These results can be considered as variations of Blind-Mani's theorem [see \textit{R. Blind} and \textit{P. Mani-Levitska}, Aequationes Math. 34, 287-297 (1987; Zbl 0634.52005)] that says that the 1-skeleton of a simple convex polytope determines its face lattice.
scientific article; zbMATH DE number 1507673