Suppose $G_0, G_1, ..., G_n$ is a sequence of graphs. Let’s call it a composition sequence iff $G_0$ has subgraphs $A_1, ... , A_n$ such that $A_i = G_i \forall i \leq n$ and their union is exactly $G_0$.
Suppose an arbitrary finite sequence of graphs is given by its length and adjacency matrices of its elements. Then it is known, that the problem of determining, whether it is a composition sequence is NP-complete.
However, what will happen if we restrict the problem to the case, when $G_0$ is known to be a complete graph? Is this particular case of the problem still NP-complete?