Flip graphs of stacked and flag triangulations of the 2-sphere

It is well-known that the flip graph of n-vertex triangulated 2-spheres is connected, i.e., each pair of n-vertex triangulated 2-spheres can be turned into each other by a sequence of edge flips for each n ! 4. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph of nvertex flag 2-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph of n-vertex stacked 2-spheres has at least as many conat most four.