On the cocircuit graph of an oriented matroid
type: Item
instance of: scholarly article
title: On the cocircuit graph of an oriented matroid
zbMATH Open document ID: 0958.52016
published in: Discrete \& Computational Geometry
publication date: 2001-04-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1507673
zbMATH Keywords: oriented matroids
zbMATH Keywords: cell complex
zbMATH Keywords: cocircuit graph
author: Raul Cordovil
author: Komei Fukuda
author: António Guedes de Oliveira
reviewed by: Stanislav Jendroľ
MaRDI profile type: Publication
full work available at URL: s004540010031
OpenAlex ID: W2008722267
DOI: 10.1007/S004540010031
description: scientific article; zbMATH DE number 1507673