The following paper presents a way to solve the maximum weight matching of a bipartite graph by reducing it to computing maximum weight matchings of two lighter bipartite graphs:
M.-Y. Kao, T. W. Lam, W.-K. Sung, and H.-F. Ting. A decomposition theorem for maximum weight bipartite matchings with applications to evolutionary trees. In Proc. of the 7th Annual European Symposium on Algorithms (ESA’99), pages 438–449, 1999.
Is there a generalization of the result which also holds for non-bipartite weighted graphs?