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Let R be a commutative ring with identity. An edge labeled graph is a graphwith edges labeled by ideals of R. A generalized spline over an edge labeledgraph is a vertex labeling by elements of R, such that the labels of any twoadjacent vertices agree modulo the label associated to the edge connectingthem. The set of generalized splines forms a subring and module over R. Such amodule it is called a generalized spline module. We show the existence of aflow-up basis for the generalized spline module on an edge labeled graph over aprincipal ideal domain by using a new method based on trails of the graph. Wealso give an algorithm to determine flow-up bases on arbitrary ordered cyclesover any principal ideal domain.

Flow-up Bases for Generalized Spline Modules on Arbitrary Graphs