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 l, then there is an n-vertex graph of treewidth at most k + 3l + 1 containing both G1 and G2 as subgraphs.