In view of the huge success of convolution neural networks (CNN) for image classification and object recognition, there have been attempts to generalize the method to general graph-structured data. One major direction is based on spectral graph theory. In this paper, we study the problem from a different perspective, by introducing parallel flow decomposition of graphs. The essential idea is to decompose a graph into families of non-intersecting one dimensional (1D) paths, after which, we may apply a 1D CNN along each family of paths. We demonstrate that the our method, which we call GFCN (graph flow convolutional network), is able to transfer CNN architectures to general graphs. We demonstrate effectiveness of the method with synthetic and real applications.