The identification of $\gamma$-rays from the predominant hadronic-background is a key aspect in their ground-based detection using Imaging Atmospheric Cherenkov Telescopes (IACTs). While current methods are limited in their ability to exploit correlations in complex data, deep learning-based models offer a promising alternative by directly leveraging image-level information. However, several challenges involving the robustness and applicability of such models remain. Designing model architectures with inductive biases relevant for the task can help mitigate the problem. Three such deep learning-based models are proposed, trained, and evaluated on simulated data: (1) a hybrid convolutional and graph neural network model (CNN-GNN) using both image and graph data; (2) an enhanced CNN-GNN variant that incorporates additional reconstructed information within the graph construction; and (3) a graph neural network (GNN) model using image moments serving as a baseline. The new combined convolution and graph-based approach demonstrates improved performance over traditional methods, and the inclusion of reconstructed information offers further potential in generalization capabilities on real observational data.