Graphs are widely used to model complicated data semantics in many application domains. In this paper, two novel and efficient algorithms Fast-ON and Fast-P are proposed for solving the subgraph isomorphism problem. The two algorithms are based on Ullman algorithm [Ullmann 1976], apply vertex-at-a-time matching manner and path-at-a-time matching manner respectively, and use effective heuristics to cut the search space. Comparing to the well-known algorithms, Fast-ON and Fast-P achieve up to 1-4 orders of magnitude speed-up for both dense and sparse graph data.