Connectivity of triangulations without degree one edges under 2-3 and   3-2 moves
  Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions), the subgraph of the Pachner graph consisting of triangulations without degree one edges is also connected, for single-vertex triangulations of closed manifolds, and ideal triangulations of manifolds with non-spherical boundary components. 
Henry Segerman
math.GT