A result on flip-graph connectivity (Q2891068)

Let \({\mathcal A}\) be a \(d\)-dimensional point configuration, that is a finite subset of \(\mathbb{R}^n\) whose affine hull has dimension \(d\). A polyhedral subdivision of \({\mathcal A}\) is a collection \(S\) of subsets of \({\mathcal A}\) so that \(\{\text{conv}(s) \mid s \in S\}\) is a polyhedral complex and \(\cup_{s\in S}\text{conv}(s)\) is exactly \(\text{conv}(A)\).NEWLINENEWLINE In the paper under review, the set \(\omega({\mathcal A})\) of all subdivisions of \(A\) has been partially ordered by the following refinement relation: a subdivision \(S\) refines another subdivision \(S'\) if every face of \(S\) is a subset of some face of \(S'\). The flip-graph of \({\mathcal A}\), denoted by \(\gamma({\mathcal A})\), is made up of the minimal and next-to-minimal elements of \(\omega({\mathcal A})\). The minimal elements in \(\omega({\mathcal A})\) are precisely the triangulations of \({\mathcal A}\). A polyhedral subdivision of a \(d\)-dimensional point configuration is \(k\)-regular if it is projected from the boundary complex of a polytope with dimension at most \(d + k\). The subgraph induced by the \(k\)-regular triangulations in the flip-graph of \({\mathcal A}\) will be denoted by \(\gamma_k({\mathcal A})\). Gel'ffand, Zelevinsky and Kapranov showed that \(\gamma_1({\mathcal A})\) is connected. In the paper under review the connectedness of \(\gamma_2({\mathcal A})\) is established.