# Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free

What is the relationship between graphs which have $$O(\log n)$$ treewidth and $$\mathcal{H}$$-minor-free graphs? Are graphs which have $$O(\log n)$$ treewidth $$\mathcal{H}$$-minor-free? I know that graphs which are $$\mathcal{H}$$-minor-free have treewidth $$O(\sqrt{n})$$ but I am concerned more with the relationship between $$O(\log n)$$ treewidth graphs and $$\mathcal{H}$$-minor-free graphs.

• What is $\mathcal H$ here? – 1.. Sep 8 '19 at 15:52
• $\mathcal{H}$ is any subgraph: math.stackexchange.com/questions/97226/h-minor-free-graph – user1246462 Sep 8 '19 at 16:54
• Take a clique on log n vertices, attach a long path to it, it is not h minor free but its treewidth is O(log n). – Saeed Sep 9 '19 at 2:55

A proper minor-closed family of graphs has bounded treewidth if and only if it's forbidden minors includes a planar graph. Thus, either the family contains all planar graphs or has bounded treewidth. In the former case the treewidth can be $$\Omega(\sqrt{n})$$ and in the latter case the treewidth is $$O(1)$$.