Undirected graph $G$ can be partitioned into several vertex blocks, each vertex pair $(u,v)$ has an edge if "$u$" and "$v$" are in the different blocks; no edge, otherwise. That is, each block pair induces a complete bipartite graph. Intuitively, if we use a vertex "$g$" to represent a block, and we add an edge $(g_i,g_j)$ if there are edges between the two blocks $i$ and $j$, then the graph $G$ is a clique.
Now, the question is: Suppose we have several such graphs $G_1,\ldots,G_n$, where a vertex $v_i$ may be in more than one $G_j$. If we combine graphs $G_1,\ldots,G_n$ by taking the union of their edge sets to get a graph $G'$, as shown in the example below, what is the name of the resulting graph? Does $G'$ have any special properties for the vertex cover problem or some other classical problems?
An example: .