  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.
