The graph of triangulations of a point configuration with \(d+4\) vertices is 3-connected
type: Item
instance of: scholarly article
title: The graph of triangulations of a point configuration with \(d+4\) vertices is 3-connected
zbMATH Open document ID: 0956.05063
author: Miguel Azaola
author: Francisco Santos
published in: Discrete \& Computational Geometry
publication date: 2001-03-04T00:00:00Z
review text: The authors study the graph bistellar flips between triangulations of a vector configuration \(\mathcal A\) with \(d+4\) elements in rank \(d+1\), as a step in the Baues problem. It is proved that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension \(d\) with \(d+4\) elements. Therefore, every pair of triangulations can be joined by a finite sequence of bistellar flips and, in the acyclic case, every triangulation has at least three geometric bistellar neighbors. In corank 4, connectivity is not known and having at least four flips is false. In corank \(2\), the results are trivial since the graph is a cycle. The authors also prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector configuration \(\mathcal B\) can be realized as a cell in some pseudo-chamber complex of \(\mathcal B\) in the same way that regular triangulations appear as cells in the usual chamber complex. All results in the paper generalize to triangulations of corank 3 oriented matroids and virtual chambers of rank 3 oriented matroids, realizable or not.
reviewed by: Ferenc Fodor
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1471112
zbMATH Keywords: point configuration
zbMATH Keywords: triangulation
zbMATH Keywords: bistellar flip
zbMATH Keywords: virtual chamber
zbMATH Keywords: pseudo-chamber complex
zbMATH Keywords: matroid
MaRDI profile type: Publication
full work available at URL: s004540010013
OpenAlex ID: W1967167259
DOI: 10.1007/S004540010013
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description: scientific article; zbMATH DE number 1471112