The treewidth and pathwidth of graph unions

For two graphs $G_1$ and $G_2$ on the same vertex set $[n]:=\{1,2, \ldots, n\}$, and a permutation $\varphi$ of $[n]$, the union of $G_1$ and $G_2$ along $\varphi$ is the graph which is the union of $G_2$ and the graph obtained from $G_1$ by renaming its vertices according to $\varphi$. We examine the behaviour of the treewidth and pathwidth of graphs under this "gluing" operation. We show that under certain conditions on $G_1$ and $G_2$, we may bound those parameters for such unions in terms of their values for the original graphs, regardless of what permutation $\varphi$ we choose. In some cases, however, this is only achievable if $\varphi$ is chosen carefully, while yet in others, it is always impossible to achieve boundedness. More specifically, among other results, we prove that if $G_1$ has treewidth $k$ and $G_2$ has pathwidth $\ell$, then they can be united into a graph of treewidth at most $k + 3 \ell + 1$. On the other hand, we show that for any natural number $c$ there exists a pair of trees $G_1$ and $G_2$ whose every union has treewidth more than $c$.