  We design a critically-sampled compact-support biorthogonal transform for
graph signals, via graph filterbanks. Instead of partitioning the nodes in two
sets so as to remove one every two nodes in the filterbank downsampling
operations, the design is based on a partition of the graph in connected
subgraphs. Coarsening is achieved by defining one "supernode" for each subgraph
and the edges for this coarsened graph derives from the connectivity between
the subgraphs. Unlike the "one every two nodes" downsampling on bipartite
graphs, this coarsening operation does not have an exact formulation in the
graph Fourier domain. Instead, we rely on the local Fourier bases of each
subgraph to define filtering operations. We apply successfully this method to
decompose graph signals, and show promising performance on compression and
denoising.
