The star-expansion of a graph

Let $H$ be a finite graph with vertex set $\{b_1,\dots ,b_k\}$. The star-expansion of $H$ is the graph $H^{\star }$ obtained by adjoining new vertices

$a_1,\dots ,a_k,v$

to $H$ and declaring the edges as follows:

1. the induced subgraph on $\{b_1,\dots ,b_k\}$ is exactly $H$;
2. for each $i\in [k]$, the tooth $a_i$ is adjacent to $b_i$;
3. the root $v$ is adjacent to every tooth $a_i$; and
4. there are no other edges incident with the new vertices.

Thus every tooth has degree $2$ unless $b_i=v$, which never happens, and the new vertices induce a star centered at $v$. When $H$ is an induced subgraph of another graph, the star-expansion is understood up to graph isomorphism in the sense of Graph isomorphisms, automorphisms and graph complements.