We got reduction graph to planar bounded treewidth graph, but this is unlikely to be true.

Let $H$, the planarizing gadget, be planar graph with four distinguished vertices $u,u',v,v'$ on the outer faces.

Take graph $G$ drawn on the plane. Add new vertex $S$, adjacent to all vertices of $G$. So far the diameter is at most two.

Replace each pair of crossing edges $(u,u'),(v,v')$ by new copy of the gadget $H$.

The resulting graph $G'$ is planar with diameter $D = 2\max(d(u,u'),d(v,v'))$ where $d$ is the distance in $H$.

The treewidth of $G'$ is $O(D)$, which is constant for fixed $H$, for reference see here.

Similar reduction with specially chosen $H$ is used to show NP-hardness of problems for planar graphs.

What is wrong with this reduction?

Correctness of the reduction is unlikely, because for bounded treewidth graphs a lot of graph invariants are computable in polynomial time and choosing suitable gadget $H$ might give relation between invariants of $G$ and $G'$, implying $P=NP$.

Another reference claims "bounded genus graphs of bounded diameter have bounded treewidth".


The diameter of $G'$ will not be bounded. Replacing edge crossings with gadgets can effectively cut each edge $O(n)$ times, so the diameter can blow up by a factor of $O(n)$.

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