  We study {\em generalized graph splines,} introduced by Gilbert, Viel, and
the last author. For a large class of rings, we characterize the graphs that
only admit constant splines. To do this, we prove that if a graph has a
particular type of cutset (e.g., a bridge), then the space of splines naturally
decomposes as a certain direct sum of submodules. As an application, we use
these results to describe splines on a triangulation studied by Zhou and Lai,
but over a different ring than they used.
