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 "2-3 Pachner graph" of such triangulations is connected. This is useful for defining invariants of a three-manifold based on the triangulation. However, there are "would-be" invariants that can only be defined on triangulations with certain properties, for example 1-efficiency or having only essential edges. Unfortunately, there are no similar connectivity results for the subgraphs of the Pachner graph with such properties. In this talk, I will describe a new connectivity result for a property implied by both 1-efficiency and essential edges: that of the triangulation having no degree one edges. Speaker(s): Henry Segerman (Oklahoma State University)