Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs

Alex Koutsoutis, Kilian Krause, Chun-Hung Liu, Mirza Redzic, Torsten Ueckerdt

TL;DR

The paper investigates the relationship between $tw(G)$, $tree\textnormal{-}α(G)$, and $tree\textnormal{-}χ(G)$ for graphs and answers a natural tree-decomposition inequality in the negative by constructing graphs via the $1$-completion operation and Burling-type graphs to separate these parameters. It introduces a strong nonexistence result: for any $f$ there is $G$ with $tw(G) > tree\textnormal{-}α(G) \cdot f(tree\textnormal{-}χ(G))$, while simultaneously establishing the universal bound $tw(G)+1 \leq tree\textnormal{-}α(G)^2 \cdot tree\textnormal{-}χ(G)$. The methods hinge on properties of $C(H)$ that preserve $tree\textnormal{-}χ$, the tight relation between $tw(C(H))$ and $|V(H)|$, and the ability to control $tree\textnormal{-}χ$ through homomorphisms and blowups. The results refine our understanding of how these tree-parameters interact and delineate the limits of joint bounds for algorithmic and structural graph theory applications.

Abstract

We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth ${\rm tw}(G)$ of a graph $G$ measures the largest number of vertices required in a bag of every tree decomposition of $G$. Similarly, the tree-independence number ${\rm tree\textnormal{-}}α(G)$ and the tree-chromatic number ${\rm tree\textnormal{-}}χ(G)$ measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of $G$. Recently, Dallard, Milanič, and Štorgel asked (JCTB, 2024) whether for all graphs $G$ it holds that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}α(G) \cdot {\rm tree\textnormal{-}}χ(G)$. We provide a negative answer for this question in a strong form: for every function $f\colon {\mathbb N} \rightarrow {\mathbb N}$, there exists a graph $G$ such that ${\rm tw}(G) > {\rm tree\textnormal{-}}α(G) \cdot f({\rm tree\textnormal{-}}χ(G))$. On the other hand, we complement this result with an upper bound, by showing that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}α(G)^2 \cdot {\rm tree\textnormal{-}}χ(G)$ for every graph $G$.

On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs

TL;DR

The paper investigates the relationship between , , and for graphs and answers a natural tree-decomposition inequality in the negative by constructing graphs via the -completion operation and Burling-type graphs to separate these parameters. It introduces a strong nonexistence result: for any there is with , while simultaneously establishing the universal bound . The methods hinge on properties of that preserve , the tight relation between and , and the ability to control through homomorphisms and blowups. The results refine our understanding of how these tree-parameters interact and delineate the limits of joint bounds for algorithmic and structural graph theory applications.

Abstract

We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth of a graph measures the largest number of vertices required in a bag of every tree decomposition of . Similarly, the tree-independence number and the tree-chromatic number measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of . Recently, Dallard, Milanič, and Štorgel asked (JCTB, 2024) whether for all graphs it holds that . We provide a negative answer for this question in a strong form: for every function , there exists a graph such that . On the other hand, we complement this result with an upper bound, by showing that for every graph .
Paper Structure (3 sections, 13 theorems, 6 equations)

This paper contains 3 sections, 13 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Completion of graphs
  3. Negative answer to \ref{['que:main-question']}

Key Result

Theorem 1

For any functions $f\colon {\mathbb R} \rightarrow {\mathbb R}_{>0}$ and $h\colon {\mathbb R} \rightarrow {\mathbb R}_{>0}$ such that $h(x)$ is non-decreasing and $h(x) = o(x\log\log x)$ as $x \to \infty$, there exists a graph $G$ such that

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9: Walczak Wal-15
  • Lemma 10
  • ...and 3 more