Let $G_k$ be the graph obtained by applying the following procedure k-times:

1. Start with a graph with single vertex $v$ (Call this graph $H$)

2. Add a vertex $u$ such that $u$ is not adjacent to any vertex of $H$ (i.e., $K:= H \cup \{u\}$) union of two graphs

3. Add a vertex $w$ such that $w$ is adjacent to all the vertices of $K$ (i.e., $J := K \vee \{w\}$) join of two graphs

4. Set $H = J$

5. Goto step 2.

My question is, is there a name for the class of graphs $\{G_k\}_{k\ge1}$? Please provide some references. Thank you.