In this paper, a new algorithm for mining frequent connected subgraphs called gRed (graph Candidate Reduction Miner) is presented. This algorithm is based on the gSpan algorithm proposed by Yan and Jan. In this method, the mining process is optimized introducing new heuristics to reduce the number of candidates. The performance of gRed is compared against two of the most popular and efficient algorithms available in the literature (gSpan and Gaston). The experimentation on real world databases shows the performance of our proposal overcoming gSpan, and achieving better performance than Gaston for low minimal support when databases are large.