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We propose novel two-channel filter banks for signals on graphs. Our designs can be applied to arbitrary graphs, given a positive semi definite variation operator, while using arbitrary vertex partitions for downsampling. The proposed generalized filter banks (GFBs) also satisfy several desirable properties including perfect reconstruction and critical sampling, while having efficient implementations. Our results generalize previous approaches that were only valid for the normalized Laplacian of bipartite graphs. Our approach is based on novel graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. These GFTs are orthogonal in an alternative inner product space which depends on the downsampling and variation operators. Our key theoretical contribution is showing that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. In addition, we study vertex domain and spectral domain properties of GFBs and illustrate their probabilistic interpretation using Gaussian graphical models. While GFBs can be defined given any choice of a vertex partition for downsampling, we propose an algorithm to optimize these partitions with a criterion that favors balanced partitions with large graph cuts, which are shown to lead to efficient and stable GFB implementations. Our numerical experiments show that partition-optimized GFBs can be implemented efficiently on 3D point clouds with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state-of-the-art approaches.