Connectivity of geometric triangulations by bistellar flips

Determine whether any two triangulations of a fixed n-vertex point set in three-dimensional Euclidean space can be connected by a sequence of geometric 2–3 and 3–2 bistellar flips.

Background

The paper relates the topological flip graph of triangulated 3-spheres to a longstanding computational-geometry problem concerning triangulations of a fixed point set in d space. Unlike the unrestricted Pachner graph, the geometric problem permits only vertex-preserving bistellar flips.

The question asks whether the geometric flip graph on triangulations of every fixed point configuration is connected. The paper cites this as an outstanding problem and uses partial results related to it as motivation for studying connectivity in the topological setting.