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?