I wonder if there is any existing method to find the clique and biclique structures in a graph that can cover all the edges in it, and every edge belongs to exactly one of the cliques or bicliques found? I try to illustrate the idea in the figure above where the red edges belong to a biclique and the blue edges belong to a clique. All the edges in the graph are covered by the structures found.
The bicliques and cliques may have overlapping nodes.
The motivation of doing this is that in social networks, the biclique structure often corresponds to an influential group of persons that reach a common group of audiences, and such structural feature cannot be described by clique ...
As noted by Tom van der Zanden, a trivial solution would be to return all the edges of the graph. I am trying to look for non trivial solutions such as those with minimum numbers of cliques and bicliques.
I wonder if anybody could point to some research in this area?