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

Vertex-disjoint Cycle Cover for graph signal processing