On cliques and bicliques

The clique graph K(G) of a a graph G is the intersection graph of the set of all (maximal) cliques of G (and K2(G)=K(K(G))). The suspension S(G) of a graph G is the graph obtained from G by adding two new vertices which are adjacent to all other vertices, but not to each other. Here, a biclique (X, Y ) is an ordered pair of not necessarily disjoint subsets of vertices of G such that each x∈X is adjacent or equal to every y∈Y and such that (X, Y ) is maximal under component-wise inclusion. Finally B(G) is the graph whose vertices are the bicliques of G with adjacencies given by (X,Y)≃(X′,Y′) if and only if X∩X′≠∅ or Y∩Y′≠∅.