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.