A novel scheme is defined for supporting service graphs to allow specific service functions to be applied to selected subsets of packet flows. According to one or more embodiments, a service graph is modeled as chains or sequences of “flow steering entities” and “service instances.” According to an embodiment, a flow steering entity (FSE) is configured to identify the flow, and select the next hop service instance in the chain. After a service instance returns the flow packet, the FSE steers the packet to the next FSE in the graph. An initial FSE is used to select matching traffic to enter the service graph. Wherever path forking and merging is required in the service graph, additional classifiers are positioned with the FSE to steer a subset of traffic along different paths of service function instances.