Decomposition of graph functions

The V -functions of Tutte [1] are generalized to U -functions on graphs with a distinguished subset of vertices. The class of U -functions of two variables generalize dichromatic polynomials as well as the W -functions defined by Tutte [2]. The values of U -functions on a graph G are characterized in terms of spanning subgraphs of G and also in terms of collections of simple graphs constructed from G . Decompositions of dichromatic polynomials as well as dichromatic U -functions are obtained in terms of decompositions of G .