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In network-on-chip (NoC), time-division-multiplexing (TDM) virtual circuits (VCs) have been proposed to satisfy the quality-of-service requirements of applications. TDM VC is a connection-oriented communication service by which two or more connections take turns to share buffers and link bandwidth using dedicated time slots. In the paper, we first give a formulation of the multinode VC configuration problem for arbitrary NoC topologies. A multinode VC allows multiple source and destination nodes on it. Then we address the two problems of path selection and slot allocation for TDM VC configuration. For the path selection, we use a backtracking algorithm to explore the path diversity, constructively searching the solution space. In the slot allocation phase, overlapped VCs must be configured such that no conflict occurs and their bandwidth requirements are satisfied. We define the concept of a logical network (LN) as an infinite set of associated (time slot, buffer) pairs with respect to a buffer on a given VC. Based on this concept, we develop and prove theorems that constitute sufficient and necessary conditions to establish conflict-free VCs. They are applicable for networks where all nodes operate with the same clock frequency but allowing different phases. Using these theorems, slot allocation for VCs is a procedure of assigning VCs to different LNs. TDM VC configuration can thus be predictable and correct-by-construction. Our experiments on synthetic and real applications validate the effectiveness and efficiency of our approach.