Reduction graph to planar bounded treewidth and bounded diameter graph 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 $O(n)$.