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The heterogeneous structure implies that a very few nodes may play the critical role in maintaining structural and functional properties of a large-scale network. Identifying these vital nodes is one of the most important tasks in network science, which allow us to better conduct successful social advertisements, immunize a network against epidemics, discover drug target candidates and essential proteins, and prevent cascading breakdowns in power grids, financial markets and ecological systems. Inspired by the nested nature of real networks, we propose a decomposition method where at each step the nodes with the lowest degree are pruned. We have strictly proved that this so-called lowest degree decomposition (LDD) is a subdivision of the famous k-core decomposition. Extensive numerical analyses on epidemic spreading, synchronization and nonlinear mutualistic dynamics show that the LDD can more accurately find out the most influential spreaders, the most efficient controllers and the most vulnerable species than k-core decomposition and other well-known indices. The present method only makes use of local topological information, and thus has high potential to become a powerful tool for network analysis.