# The treewidth and pathwidth of graph unions

For two graphs G_1 and G_2 on the same vertex set [n]:={1,2, …, n}, and a permutation φ of [n], the union of G_1 and G_2 along φ is the graph which is the union of G_2 and the graph obtained from G_1 by renaming its vertices according to φ. 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 φ we choose. In some cases, however, this is only achievable if φ 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 ℓ, then they can be united into a graph of treewidth at most k + 3 ℓ + 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.

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