Let $S$ be a finite sequence of positive integers of length $n$. Let $G$ be a finite graph containing $n$ not null edgeless subgraphs. Call these subgraphs $G_0,G_1,\ldots,G_{n-1}$ and let each $G_i$ have order $S_i$. Define the edges of $G$ such that, for all $i>0$, $G_i$ forms a complete bipartite graph with $G_{i-1}$. $G$ contains no other edges.
Does this graph have a name? I would appreciate any suggestions for a name if none already exists.