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We use circulant structures to present a new framework for multiresolution analysis and processing of graph signals. Among the essential features of circulant graphs is that they accommodate fundamental signal processing operations, such as linear shift-invariant filtering, downsampling, upsampling, and reconstruction-features that offer substantial advantage. We design two-channel, critically-sampled, perfect-reconstruction, orthogonal lattice-filter structures to process signals defined on circulant graphs. To extend our reach to noncirculant graphs, we present a method to decompose a connected, undirected graph into a combination of circulant graphs. To evaluate our proposed framework, we offer examples of synthetic and real-world graph signal data and their multiscale decompositions.