Eigenvectors of the Laplacian of a cycle graph exhibit the sinusoidal characteristics of the standard DFT basis, and signals defined on such graphs are amenable to linear shift invariant (LSI) operations. In this paper we propose to reduce a generic graph to its vertex-disjoint cycle cover, i.e., a set of subgraphs that are cycles, that together contain all vertices of the graph, and no two subgraphs have any vertices in common. Additionally if the weight of an edge in the graph is a function of the variation in the signals on its vertices, then maximally smooth cycles can be found, such that the resulting DFT does not have high frequency components. We show that an image graph can be reduced to such low-frequency cycles, and use that to propose a simple image denoising algorithm.