Name of an inductively defined sequence of graphs

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.

• Is it the join of $\overline{K_{k+1}}$ and $K_k$? I doubt if there is a name. Commented Sep 15, 2021 at 1:45
• @BrendanMcKay, no. The initial vertex $v$ from step 1 and the $w$ vertices form $K_{k+1}$, and the $u$ vertices have degrees $1$ to $k$ because (indexing by iteration at which they're added) $u_i - w_j$ iff $i \le j$. Commented Sep 15, 2021 at 13:50
• @PeterTaylor Right, my mistake. I still doubt if there's a name. Commented Sep 15, 2021 at 13:59

As David Eppstein remarked, those graphs will all be threshold graphs. Moreover, they will be universal threshold graphs, in the sense that $$G_k$$ contains all threshold graphs on $$k+1$$ vertices. This is also discussed in Section 4.1.6 of Michael Engen's thesis.