The treewidth and pathwidth of graph unions

TL;DR

This work addresses whether two $n$-vertex graphs of bounded treewidth can be combined into another $n$-vertex graph of bounded treewidth that contains both as subgraphs. It delivers a strong negative result by showing that there exist $n$-vertex trees whose any gluing has treewidth arbitrarily large, even when gluing binary and ternary trees, using a number-theoretic style argument. On the positive side, the paper proves that if one graph has treewidth $k$ and the other pathwidth $l$, a gluing with treewidth at most $k+3l+1$ exists, with sharper bounds in special cases such as vertex covers and caterpillars; it also develops a framework based on smooth tree decompositions and tilts. These results delineate the boundary between gluing that preserves bounded treewidth and gluing that inevitably yields large width, with implications for multilayer network design and graph class theory, and they pose open questions about gluing more than two components and planarity constraints.

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.

Figures (3)

• Figure 1: The union of two trees along different permutations.
• Figure 2: Balanced binary and ternary trees
• Figure 3: An illustration of two trees and their tilts

Theorems & Definitions (33)

• Theorem 0
• Theorem 0
• Lemma 1
• Lemma 2
• Theorem 3
• proof
• Theorem 4: Kinnersley K92
• Lemma 5
• proof
• Lemma 6