Constrained paths in the flip-graph of regular triangulations
This note regards some special geometric operation that arises in many applications from mesh generation to algebraic geometry. A triangulation of a point set \(A\) is a simplicial complex \(T\) with a geometric realization \(| T |\) such that the (1) vertices of the simplices are points of \(A\), (2) the usual abstract properties of simplicial complexes are respected in the embedding, and (3) \(| T |\), as a subset of Euclidean space, is precisely the convex hull of \(A\). Geometric bistellar flips (also called geometric Pachner moves) are simple geometric operations that replace a subcomplex inside the triangulation by a new simplicial complex that yields a new triangulation. Note that the set of vertices \(A\) is fixed and thus the number of distinct triangulations is finite. A triangulation is regular when it is the projection of a convex triangulated surface of one higher dimension. A classical example is a Delaunay triangulation. It has been known for some time now that the set of all regular triangulations is connected by flips (a result going back to 1990 due to Gel'fand Kapranov and Zelevinsky). The present note shows that in fact any pair of regular triangulations is connected by a sequence of finitely many flips where none of their common faces are destroyed. As a consequence it follows the connectivity of the subgraph induced by all regular triangulations that share the same vertex set.
scientific article; zbMATH DE number 5144508