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This paper presents a novel deep neural network framework for RGB-D salient object detection by controlling the message passing between the RGB images and depth maps on the feature level and exploring the long-range semantic contexts and geometric information on both RGB and depth features to infer salient objects. To achieve this, we formulate a dynamic message propagation (DMP) module with the graph neural networks and deformable convolutions to dynamically learn the context information and to automatically predict filter weights and affinity matrices for message propagation control. We further embed this module into a Siamese-based network to process the RGB image and depth map respectively and design a multi-level feature fusion (MFF) module to explore the cross-level information between the refined RGB and depth features. Compared with 17 state-of-the-art methods on six benchmark datasets for RGB-D salient object detection, experimental results show that our method outperforms all the others, both quantitatively and visually.