The graph edit distance (GED) is a well-established distance measure widely used in many applications. However, existing methods for the GED computation suffer from several drawbacks including oversized search space, huge memory consumption, and lots of expensive backtracking. In this paper, we present BSS_GED, a novel vertex-based mapping method for the GED computation. First, we create a small search space by reducing the number of invalid and redundant mappings involved in the GED computation. Then, we utilize beam-stack search combined with two heuristics to efficiently compute GED, achieving a flexible trade-off between available memory and expensive backtracking. Extensive experiments demonstrate that BSS GED is highly efficient for the GED computation on sparse as well as dense graphs and outperforms the state-of-the-art GED methods. In addition, we also apply BSS_GED to the graph similarity search problem and the practical results confirm its efficiency.