  The degree sequence optimization problem is to find a subgraph of a given
graph which maximizes the sum of given functions evaluated at the subgraph
degrees. Here we study this problem by replacing degree sequences, via suitable
nonlinear transformations, by suitable degree enumerators, and we introduce
suitable degree enumerator polytopes.
  We characterize their vertices, that is, the extremal degree enumerators, for
complete graphs and some complete bipartite graphs, and use these
characterizations to obtain simpler and faster algorithms for optimization over
degree sequences for such graphs.
