Is it possible to unite two $n$-vertex trees such that the resulting graph has bounded tree-width?


Formally, does there exists a constant $k$ such that given two $n$-vertex trees $T\_1$ and $T\_2$ there exist labeled trees $T\_1'=([n], E\_1)$ and $T\_2' =([n],E\_2)$ with $T\_1' \simeq T\_1$, $T\_2' \simeq T\_2$ and
$$
\texttt{treewidth} (T\_1' \cup T\_2') \leq k,
$$
where $T\_1' \cup T\_2' = ([n], E\_1 \cup E\_2)$.


**Special cases**


1. $T\_1 \simeq P\_n$ and $T\_2 \simeq P\_n$. While one can 'construct' a grid out of two large paths (and a grid has 'large' tree-width), $T\_1$ and $T\_2$ can be united into a path $P\_n$ (which has treewidth 1).
2. $T\_1$ is a star, $T\_2$ is an arbitrary tree. In this case, no matter how we unite the trees we get a chordal graph with maximum clique size 3, and therefore of tree-width 2.
3. It is also possible to unite a path with an arbitrary tree into a graph of bounded tree-width. But it is not clear if this is always possible for a pair of arbitrary trees.



Tree-width of a union of two trees
It is shown in [Bogdan Alecu, Vadim Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev: "The treewidth and pathwidth of graph unions"](https://arxiv.org/abs/2202.07752) that the general question has a negative answer.


