I was wondering if the following decomposition of graphs has been studied, whether it has a name, and what the literature might be on it.

Given a labelled graph *G*, we decompose its edge-set as a symmetric difference of complete bipartite graphs, where each bipartition may contain any vertices of *G* that we wish (under the constraint of being disjoint of course), and where each such bipartite graph (each "term" in the decomposition) may be of a different size. We may ask for a "minimal" such decomposition, by different ways of assessing the importance of the "terms": each term has equal weight, or a weight equal to the number of vertices that it contains, and we ask for the decomposition of the smallest total weight.

Has such a decomposition been studied before?

(a)the $K_{4,4}$ graph on the bipartitions {a,b,c,d} and {e,f,g,h}, and(b)the isolated edges (copies of $K_{1,1}$) ab and de. Giving each $K_{m,n}$ term in a decomposition a total weight of $(m+n)$, the decomposition of $G$ consisting of the symmetric "difference" of each edge individually is 32 (twice the number of edges), while the second decomposition is $(4+4) + (1+1) + (1+1) = 12$. $\endgroup$ – Niel de Beaudrap May 25 '12 at 13:51