TITLE: A graph parameter possibly related to treewidth
QUESTION [10 upvotes]: (Cross-posted from the Theoretical Computer Science StackExchange site after there was no conclusive answer after a week.)
I am interested in graphs on $n$ vertices which can be produced via the following process.

Start with an arbitrary graph $G$ on $k\le n$ vertices. Label all the vertices in $G$ as unused.
Produce a new graph $G'$ by adding a new vertex $v$, which is connected to one or more  unused vertices in $G$, and is not connected to any used vertices in $G$. Label $v$ as unused.
Label one of the vertices in $G'$ to which $v$ is connected as used.
Set $G$ to $G'$ and repeat from step 2 until $G$ contains $n$ vertices.

Call such graphs "graphs of complexity $k$" (apologies for the vague terminology). For example, if $G$ is a graph of complexity 1, $G$ is a path.
I would like to know if this process has been studied before. In particular, for arbitrary $k$, is it NP-complete to determine whether a graph has complexity $k$?
This problem appears somewhat similar to the question of whether $G$ is a partial $k$-tree, i.e. has treewidth $k$. It is known that determining whether $G$ has treewidth $k$ is NP-complete. However, some graphs (stars, for example) may have much smaller treewidth than the measure of complexity discussed here.

REPLY [4 votes]: Following Felix's suggestion, and corroborating Ashley's characterization,
here are five examples of graphs grown from
$K_2$, i.e., $E=\lbrace (1,2) \rbrace$:

     


The vertices are numbered in the order in which they were added.

Here is just one more, this time grown from $K_5$, to give a gestalt view of the influence of
the start $G$: