The treewidth and pathwidth of graph unions

Given two $n$-vertex graphs $G1$ and $G2$ of bounded treewidth, is there an $n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G1$ and $G2$? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if $G1$ is a binary tree and $G2$ is a ternary tree. We also provide an extensive study of cases where such `gluing' is possible. In particular, we prove that if $G1$ has treewidth $k$ and $G2$ has pathwidth $\ell$, then there is an $n$-vertex graph of treewidth at most $k + 3 \ell + 1$ containing both $G1$ and $G2$ as subgraphs.