The graph of triangulations of a point configuration with \(d+4\) vertices is 3-connected
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.
scientific article; zbMATH DE number 1471112