Let $\mathcal{F}$ be the smallest family of (simple, finite) graphs such that:


1. The empty graph is in $\mathcal{F}$.
2. If $G \in \mathcal{F}$, then $G\_{\bullet}$, the graph obtained from $G$ by adding another vertex $v$ connected by an edge to each vertex in $G$, is also in $\mathcal{F}$.
3. If $G,H \in \mathcal{F}$, then $G \cup H \in \mathcal{F}$.


There are natural bijections between $\{G \in \mathcal{F}: \textrm{$G$ has $n$ vertices}\}$ and the set of rooted (but unlabeled) forests on $n$ vertices and between $\{G \in \mathcal{F}: \textrm{$G$ has $n$ vertices and is connected}\}$ and the set of rooted (but unlabeled) trees on $n$ vertices.


Does $\mathcal{F}$ have a name? I want to call it something like the set of "generalized threshold graphs" because the rules that define it are close to those of [threshold graphs](http://en.wikipedia.org/wiki/Threshold_graph).



Does the family of graphs closed under taking unions and adding dominating vertices have a name?
It appears that you are looking at the family of [*quasi-threshold graphs*](http://www.graphclasses.org/classes/gc_781.html). These are the graphs which can be built, starting from $K\_1$, by repeatedly taking the disjoint union of two quasi-threshold graphs, or the join of a quasi-threshold graph with a single new vertex.


By the way, if you relax your conditions a bit more, you get the extremely well-studied family of [*cographs*](http://en.wikipedia.org/wiki/Cograph). These are the graphs which can be built, starting from $K\_1$, by repeatedly taking the disjoint union or join of two smaller cographs.


