Graphs are considered to be one of the best studied data structures in discrete mathematics and computer science. Hence, data mining on graphs has become quite popular in the past few years. The problem of finding frequent itemsets in conventional data mining on transactional databases, thus transformed to the discovery of subgraphs that frequently occur in the graph dataset containing either single graph or multiple graphs. Most of the existing algorithms in the field of frequent subgraph discovery adopts an Apriori approach based on generation of candidate set and test approach. The problem with this approach is the costlier candidate set generation, particularly when there exist more number of large subgraphs. The research goals in frequent subgraph discovery are to evolve (i) mechanisms that can effectively generate candidate subgraphs excluding duplicates and (ii) mechanisms that find best processing techniques that generate only necessary candidate subgraphs in order to discover the useful and desired frequent subgraphs. In this paper, a two phase approach is proposed by integrating Apriori algorithm on graphs to frequent subgraph (FS) tree to discover frequent subgraphs in graph datasets.