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.



Name of an inductively defined sequence of graphs
It's not quite the same question, but the graphs that can be obtained by repeating either of the two operations (add a disjoint vertex or a dominating vertex), not necessarily in strict alternation, are called [threshold graphs](https://en.wikipedia.org/wiki/Threshold_graph).


