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Deterministic routing has emerged as a promising technology for future non-terrestrial networks (NTNs), offering the potential to enhance service performance and optimize resource utilization. However, the dynamic nature of network topology and resources poses challenges in establishing deterministic routing. These challenges encompass the intricacy of jointly scheduling transmission links and cycles, as well as the difficulty of maintaining stable end-to-end (E2E) routing paths. To tackle these challenges, our work introduces an efficient temporal graph-based deterministic routing strategy. Initially, we utilize a time-expanded graph (TEG) to represent the heterogeneous resources of an NTN in a time-slotted manner. With TEG, we meticulously define each necessary constraint and formulate the deterministic routing problem. Subsequently, we transform this nonlinear problem equivalently into solvable integer linear programming (ILP), providing a robust yet time-consuming performance upper bound. To address the considered problem with reduced complexity, we extend TEG by introducing virtual nodes and edges. This extension facilitates a uniform representation of heterogeneous network resources and traffic transmission requirements. Consequently, we propose a polynomial-time complexity algorithm, enabling the dynamic selection of optimal transmission links and cycles on a hop-by-hop basis. Simulation results validate that the proposed algorithm yields significant performance gains in traffic acceptance, justifying its additional complexity compared to existing routing strategies.