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Tree decompositions whose trees are subgraphs: An application of Simon's factorization

Romain Bourneuf, Gwenaël Joret, Piotr Micek, Martin Milanič, Michał Pilipczuk

Abstract

We show that every connected graph $G$ has a tree decomposition indexed by a tree $T$ such that $T$ is a subgraph of $G$ and the width of the tree decomposition is bounded from above by a function of the pathwidth of $G$. This answers a question of Blanco, Cook, Hatzel, Hilaire, Illingworth, and McCarty (2024), who proved that it is not possible to have such a tree decomposition whose width is bounded by a function of the treewidth of $G$. The proof relies on Simon's Factorization Theorem for finite semigroups, a tool that has already been applied successfully in various areas of graph theory and combinatorics in recent years. Our application is particularly simple and can serve as a good introduction to this technique.

Tree decompositions whose trees are subgraphs: An application of Simon's factorization

Abstract

We show that every connected graph has a tree decomposition indexed by a tree such that is a subgraph of and the width of the tree decomposition is bounded from above by a function of the pathwidth of . This answers a question of Blanco, Cook, Hatzel, Hilaire, Illingworth, and McCarty (2024), who proved that it is not possible to have such a tree decomposition whose width is bounded by a function of the treewidth of . The proof relies on Simon's Factorization Theorem for finite semigroups, a tool that has already been applied successfully in various areas of graph theory and combinatorics in recent years. Our application is particularly simple and can serve as a good introduction to this technique.
Paper Structure (10 sections, 9 theorems, 38 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. k-interface graphs and abstractions
  4. Simon's Factorization Theorem
  5. Proof of main result
  6. Forest decompositions
  7. Intuition and proof sketch
  8. The proof
  9. + is associative
  10. [[.]] is a semigroup homomorphism

Key Result

Theorem 0

There exists a function $f:\mathbb{N} \to \mathbb{N}$ such that, for every connected graph $G$ of pathwidth less than $k$, there is a tree decomposition of $G$ of width at most $f(k)$ indexed by a tree $T$ that is a spanning tree of $G$. Moreover, in this tree decomposition, every vertex belongs to

Figures (1)

  • Figure 1: Left: The graph $G$ for $n=4$. Right: A possible forest $F_{10}$ produced by the algorithm. Observe that after connecting $v_4$ to $F_{10}$ to create $F_{11}$, either $u_4$ or $w_4$ will have to be added to the bag of $v_0$ in order to be added to the bag of $v_4$.

Theorems & Definitions (25)

  • Theorem 0
  • Lemma 1
  • Theorem 2: Simon's Factorization Theorem S95K08
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 15 more