In
^{2} the flip graph is connected, providing a
basis for algorithms to iterate toward the Delaunay triangulation.
A decade ago, several [EPW90,Joe91] asked whether the
same holds in
^{3} (when no four points
are coplanar), but the question remains unresolved.
It is not even known whether the flip graph might contain an
isolated node.
Settled in the negative for points in
^{5}
by Santos [San00],
by constructing polytopes with a space of triangulations
not connected via bistellar flips.
Settled in the negative for topological tetrahedralizations in 3D,
but the constructed tetrahedralization cannot be realized geometrically
[DFM04].
Settled in the positive for flip graphs of regular triangulations
in any dimension in [PL07],
based on earlier work of Gelfand, Kapranov and Zelevinsky.
The result in [PL07] connects by flips that neither remove
nor add vertices (i.e., 2-to-3 or 3-to-2 flips in 3D),
whereas the earlier work by Gelfand et al. permits all flips
(e.g., 1-to-4 and 4-to-1 flips in 3D).