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This paper develops the theoretical background for the design of deadlock-free adaptive routing algorithms for virtual cut-through and store-and-forward switching. This theory is valid for networks using either central buffers or edge buffers. Some basic definitions and three theorems are proposed, developing conditions to verify that an adaptive algorithm is deadlock-free, even when there are cyclic dependencies between routing resources. Moreover, we propose a necessary and sufficient condition for deadlock-free routing. Also, a design methodology is proposed. It supplies fully adaptive, minimal and non-minimal routing algorithms, guaranteeing that they are deadlock-free.The theory proposed in this paper extends the necessary and sufficient condition for wormhole switching previously proposed by us. The resulting routing algorithms are more flexible than the ones for wormhole switching. Also, the design methodology is much easier to apply because it automatically supplies deadlock-free routing algorithms.