The treewidth is a parameter of the graph that describes its similarity to a tree. Treewidth is NP-hard to find. For the introduction please see [wikipedia](https://en.wikipedia.org/wiki/Treewidth)


The question is how to generate interesting graphs with a specified treewidth? I know that:
1) Any graph containing a $k+1$-clique has treewidth at least $k$
2) Any graph containing a $k \times k$ grid has treewidth at least $k$


Are other variants possible?



Is there a way to generate a graph of specified treewidth
This is an extended comment. Here are some classes of graphs that should be interesting:


* $k$-trees by definition have treewidth $k$; moreover, adding any edge to a $k$-tree increases treewidth. This should be an interesting example since $k$-trees are "borderline maximal" with treewidth $k$.
* Graphs with treewdith $\leq k$ are [famously characterized](https://en.wikipedia.org/wiki/Treewidth#Forbidden_minors) by a finite set of forbidden minors. Each of these minors is again interesting since they are "borderline minimal" with treewidth $> k$. Again, deleting any edge from any such critical graph decreases treewidth.
* Edge subdivision does not change treewidth, hence any interesting graph can be artificially "inflated" to produce another, sparser one.


