Graph edit distance is an error tolerant matching technique emerged as a powerful and flexible graph matching paradigm that can be used to address different tasks in pattern recognition, machine learning and data mining; it represents the minimum-cost sequence of basic edit operations to transform one graph into another by means of insertion, deletion and substitution of vertices and/or edges. A widely used method for exact graph edit distance computation is based on the A* algorithm. To overcome its high memory load while traversing the search tree for storing pending solutions to be explored, we propose a depth-first graph edit distance algorithm which requires less memory and searching time. An evaluation of all possible solutions is performed without explicitly enumerating them all. Candidates are discarded using an upper and lower bounds strategy. A solid experimental study is proposed; experiments on a publicly available database empirically demonstrated that our approach is better than the A* graph edit distance computation in terms of speed, accuracy and classification rate.