The separation of the control and data planes in Software Defined Network (SDN) enables its widespread application in large-scale network scenarios such as data centers, the Internet of Things (IoT), and cloud networks. However, this decoupled network architecture exposes the network to saturation attacks. Detecting saturation attacks based on Graph Neural Network (GNN) is a popular research topic in SDN. Nevertheless, the commonly used k-Nearest Neighbors (k-NN) graph in GNN overlooks short-term flow features, failing to effectively aggregate node information and preventing the model from fully leveraging the temporal characteristics of flows. To enhance the accuracy of saturation attack detection by utilizing both long- and short-term flow features, this study proposes a saturation attack detection method called HGNM, based on long-short-term flow graphs and a hybrid GNN. This method collects long- and short-term flow features by setting two sampling times. Additionally, this study designs a long-short-term flow graph generation method, named LSGH, based on the gray relational coefficient to construct long-short-term flow graphs, ensuring that the flow graphs encompass all features of the flows. The study also devises a hybrid GNN model, GU-GCN, by paralleling the GRU and GCN to capture both the temporal and spatial features of the flows, thereby improving the model's accuracy in detecting saturation attacks. Experimental results demonstrate that, on the generated graphs, the LSGH method outperforms the k-NN and CRAM algorithms in effectively enhancing the detection accuracy of the model. Moreover, compared to the other models, the GU-GCN model exhibits performance improvements in terms of accuracy, precision, recall, F1-score, ROC curve, PR curve, and confusion matrix.