  Generalized splines on a graph $G$ with edge labels in a commutative ring $R$
are vertex labelings such that if two vertices share an edge in $G$, the
difference between the vertex labels lies in the ideal generated by the edge
label. When $R$ is an integral domain, the set of all such splines is a
finitely generated $R$-module $R_G$ of rank $n$, the number of vertices of $G$.
We find determinantal conditions on subsets of $R_G$ that determine whether
$R_G$ is a free module, and if so, whether a so called "flow-up class basis"
exists.
