Recently, numerous graph neural network methods have been developed to tackle domain shifts in graph data. However, these methods presuppose that unlabeled target graphs belong to categories previously seen in the source domain. This assumption could not hold true for in-the-wild target graphs. In this paper, we delve deeper to explore a more realistic problem open-set graph domain adaptation. Our objective is to not only identify target graphs from new categories but also accurately classify remaining target graphs into their respective categories under domain shift and label scarcity. To address this challenging problem, we introduce a novel method named Dual Structured Exploration with Mixup (DREAM). DREAM incorporates a graph-level representation learning branch as well as a subgraph-enhanced branch, which jointly explores graph topological structures from both global and local viewpoints. To maximize the use of unlabeled target graphs, we train these two branches simultaneously using posterior regularization to enhance their inter-module consistency. To accommodate the open-set setting, we amalgamate dissimilar samples to generate virtual unknown samples belonging to novel classes. Moreover, to alleviate domain shift, we establish a k nearest neighbor-based graph-of-graphs and blend multiple neighbors of each sample to produce cross-domain virtual samples for inter-domain consistency learning. Extensive experiments validate the effectiveness of our proposed DREAM compared with various state-of-the-art approaches in different settings.