There are several interesting classes of graphs with bounded treewidth. For instance, trees (treewidth 1), series parallel graphs (treewidth 2), outerplanar graphs (treewidth 2), $k$-outerplanar graphs (treewidth O(k)), graphs of branchwidth $k$ (treewidth O(k)),...
Question: Are there examples of interesting classes of graphs whose treewidth is not bounded by a constant, but by a low-growing function?
- Are there well known graph classes with treewidth $O(\log\log n)$?
- Are there well known graph classes with treewidth $O(\log n)$?
I would also be interested in classes of graphs with treewidth $O(\log^k n)$ or $O(\log\log...n)$ where the logarithm is repeated a constant number of times.
Obs: Of course it is easy to cook up artificial families of graphs with a given treewidth, like the family of $\;O(\log n)\times n\;$ grids. So I'm primarily looking for family of graphs which have been studied in other branches of graph theory and which happen to have treewidth $O(\log n)$ or $O(\log\log n)$, but non-constant treewidth.