Currently, network applications, such as audio, video, and augmented reality, have different stringent service requirements. They require service provision through end-to-end connections via other networks with different operating environments or service conditions. Therefore, network operators require information on their own and other networks to provide end-to-end services traversing several networks while guaranteeing their quality of service (QoS) requirements. This study proposes an inter-domain flow decision method that satisfies QoS requirements using a directed acyclic graph (DAG) in multi-domain and hierarchical software defined networking (SDN) networks. There are multiple local networks with SDN controllers that are connected to the global SDN controller. The flow decision in the proposed method is based on the effective bandwidth theory of the martingale process. The effectiveness of the proposed method is demonstrated by comparing it with existing SDN-based path selection methods using the Riverbed Modeler (older, OPNET) and OpenDaylight SDN controllers.