A result on flip - graph connectivity

A polyhedral subdivision of a d-dimensional point configuration A is k-regular if it is projected from the boundary complex of a polytope with dimension at most d+k. Call γk(A) the subgraph induced by k-regular triangulations in the flip-graph of A. Gel’fand, Kapranov, and Zelevinsky have shown that γ1(A) is connected. It is established here that γ2(A) is connected as well. 1. Flip-graph connectivity and regular subdivisions Let A be a d-dimensional point configuration, that is a finite subset of R whose affine hull has dimension d. A polyhedral subdivision of A is a collection S of subsets of A so that {conv(s) : s ∈ S} is a polyhedral complex and ∪s∈Sconv(s) is exactly conv(A). In this paper, the set ω(A) of all subdivisions of A will be 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′. A first remarkable subposet of ω(A) is the flip-graph of A, made up of the minimal and next-to-minimal elements of ω(A), and denoted by γ(A) in the following. Observe that the minimal elements in ω(A) are precisely the triangulations of A. Its next-to-minimal elements will be called flips according to the definition found in [19] (definition 4.7.) and [21] (remark 1.17.). This definition identifies flips within the refinement poset of a point configuration, but other equivalent definitions are possible [18,19,21]. In particular, it is more usual to define flips not as subdivisions, but as local operations instead that transform a triangulation of a point configuration into another one. The flip-graph of a point configuration A then is the graph whose vertices are the triangulations of A and whose edges are its flips. While this definition is equivalent to the one above [18,21], it better shows that flips are also used to build triangulations incrementally [8,12]. The connectivity of the flip-graph then arises as a natural question: is it always possible to transform a triangulation of a point configuration into Mathematics Institute, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. Present address: EFREI, 30-32 avenue de la République, 94800 Villejuif, France e-mail: [email protected]