Earlier work has developed the underpinnings of a theory of scheduling computations having intertask dependencies - modeled via dags - for Internet-based computing. The goal of the schedules produced is to render tasks eligible for execution at the maximum possible rate. This goal aims: (a) to utilize remote clients? computational resources well, by always having work to allocate to an available client; (b) to lessen the likelihood of the "gridlock" that ensues when a computation stalls for lack of eligible tasks. The dags handled by the theory thus far are those that can be constructed from a given collection of bipartite building-block dags via the operation of dagcomposition. The current paper extends the range of applicability of the theory by significantly expanding the repertoire of building-block dags that the scheduling algorithms can handle. Thereby, the theory can now schedule large classes of "expansive" and "reductive" dags optimally.