So, if you have two graphs (both with a designated edge), you could take the direct sum of those two graphs, where you glue those two graphs to each other along the designated edge. I think different literature use different definitions. Anyways, see the picture below as an example of gluing two cycles to each other of length 6 along some edge.
I was wondering if there is anything known about what happens to the tree width of the new graph, given the tree widths of the original two graphs? Is there some literature about it. I thought perhaps that $tw(G \oplus H) = \max({tw(G), tw(H)})$, where $\oplus$ denotes the operation I described above, and $tw$ being the tree width. I'm not even sure if this is true, and if it'd be true, where I could read more about this, or whether there are known proofs I could refer to. If not, maybe something is known about tree widths of graphs under different kind of operations. If something is unclear about my question, let me know. Thank you!