Universality theorems for generalized splines

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating set'' for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.