The treewidth and pathwidth of graph unions

TL;DR

This paper investigates whether two graphs of bounded treewidth can be combined into a larger graph of bounded treewidth containing both as subgraphs, providing negative results and bounds for specific cases.

Contribution

It proves that certain combinations of bounded treewidth graphs cannot always be embedded into a single bounded treewidth graph, and establishes bounds for specific graph parameters.

Findings

Negative answer for union of binary and ternary trees

Bound on treewidth when combining graphs with different parameters

Extensive study of conditions for graph union embeddings

Abstract

Given two $n$-vertex graphs $G_1$ and $G_2$ of bounded treewidth, is there an $n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G_1$ and $G_2$? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if $G_1$ is a binary tree and $G_2$ is a ternary tree. We also provide an extensive study of cases where such `gluing' is possible. In particular, we prove that if $G_1$ has treewidth $k$ and $G_2$ has pathwidth $\ell$, then there is an $n$-vertex graph of treewidth at most $k + 3 \ell + 1$ containing both $G_1$ and $G_2$ as subgraphs.