What is the terminology for assigning $K_{m_i}$ (complete graph) to the $i$ th vertex, 'joining' if the corresponding vertices are adjacent?

Given a connected graph $$G$$ with $$n$$ vertices and given set of $$\{m_1,m_2,...,m_n\}$$ $$n$$ integers, we form a new graph $$G{'}$$ by considering the complete graph $$K_{m_i}$$ for each vertex i and 'join' (in the sense of graph theory) two of such complete graphs if the corresponding vertices are adjacent. Is there a name for this graph $$G{'}$$ associated to the Graph $$G$$?

By joining of two graphs $$G_1$$ and $$G_2$$, I mean introducing edges from all the vertices of $$G_1$$ to all the vertices of $$G_2$$ and vice versa, keeping the original edges as is.

• I think that your construction is called the lexicographic product of graphs, where in your case $G^ := G \cdot K_m$. – TheNicanova Feb 12 '16 at 4:01