Open-set single-source domain generalization aims to use a single-source domain to learn a robust model that can be generalized to unknown target domains with both domain shifts and label shifts. The scarcity of the source domain and the unknown data distribution of the target domain pose a great challenge for domain-invariant feature learning and unknown class recognition. In this paper, we propose a novel learning approach based on domain expansion and boundary growth to expand the scarce source samples and enlarge the boundaries across the known classes that indirectly broaden the boundary between the known and unknown classes. Specifically, we achieve domain expansion by employing both background suppression and style augmentation on the source data to synthesize new samples. Then we force the model to distill consistent knowledge from the synthesized samples so that the model can learn domain-invariant information. Furthermore, we realize boundary growth across classes by using edge maps as an additional modality of samples when training multi-binary classifiers. In this way, it enlarges the boundary between the inliers and outliers, and consequently improves the unknown class recognition during open-set generalization. Extensive experiments show that our approach can achieve significant improvements and reach state-of-the-art performance on several cross-domain image classification datasets.