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Moving Object Detection (MOD) is a fundamental step for many computer vision applications. MOD becomes very challenging when a video sequence captured from a static or moving camera suffers from the challenges: camouflage, shadow, dynamic backgrounds, and lighting variations, to name a few. Deep learning methods have been successfully applied to address MOD with competitive performance. However, in order to handle the overfitting problem, deep learning methods require a large amount of labeled data which is a laborious task as exhaustive annotations are always not available. Moreover, some MOD deep learning methods show performance degradation in the presence of unseen video sequences because the testing and training splits of the same sequences are involved during the network learning process. In this work, we pose the problem of MOD as a node classification problem using Graph Convolutional Neural Networks (GCNNs). Our algorithm, dubbed as GraphMOD-Net, encompasses instance segmentation, background initialization, feature extraction, and graph construction. GraphMOD-Net is tested on unseen videos and outperforms state-of-the-art methods in unsupervised, semi-supervised, and supervised learning in several challenges of the Change Detection 2014 (CDNet2014) and UCSD background subtraction datasets.