For semantic segmentation of remote sensing images (RSI), trade-off between representation power and location accuracy is quite important. How to get the trade-off effectively is an open question, where current approaches of utilizing attention schemes or very deep models result in complex models with large memory consumption. Compared with the popularly-used convolutional neural network (CNN) with fixed square kernels, graph convolutional network (GCN) can explicitly utilize correlations between adjacent land covers and conduct flexible convolution on arbitrarily irregular image regions. However, the problems of large variations of target scales and blurred boundary cannot be easily solved by GCN, while densely connected atrous convolution network (DenseAtrousCNet) with multi-scale atrous convolution can expand the receptive fields and obtain image global information. Inspired by the advantages of both GCN and Atrous CNN, a two-stream deep neural network for semantic segmentation of RSI (RSI-Net) is proposed in this paper to obtain improved performance through modeling and propagating spatial contextual structure effectively and a novel decoding scheme with image-level and graph-level combination. Extensive experiments are implemented on the Vaihingen, Potsdam and Gaofen RSI datasets, where the comparison results demonstrate the superior performance of RSI-Net in terms of overall accuracy, F1 score and kappa coefficient when compared with six state-of-the-art RSI semantic segmentation methods.