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.

  • $\begingroup$ What is $\mathcal H$ here? $\endgroup$ – T.... Sep 8 '19 at 15:52
  • $\begingroup$ $\mathcal{H}$ is any subgraph: math.stackexchange.com/questions/97226/h-minor-free-graph $\endgroup$ – user1246462 Sep 8 '19 at 16:54
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    $\begingroup$ 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). $\endgroup$ – 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 $O(1)$.

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