Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On treewidth and maximum cliques

Maria Chudnovsky, Nicolas Trotignon

TL;DR

The paper investigates the relationship between treewidth and clique number in hereditary graph classes by introducing $(f,\ell)$-layered wheels, a flexible infinite construction that enforces holes of length at least $\ell$ while controlling clique growth across layers via a slow function $f$. It proves a general width bound: every finite induced subgraph $H$ has $\mathrm{tw}(H)$ bounded by a function of $\omega(H)$, yet the dependence can be tuned to be arbitrarily large, enabling a range of counterexamples. The authors then leverage this to disprove conjectures about tree-independence numbers and to challenge Hajebi’s proposals on $K_c$-free obstructions, by producing graphs with large treewidth whose induced subgraphs avoid large-width configurations. The results establish a versatile toolkit for engineering width properties in hereditary classes and illuminate when width notions align or diverge, with implications for algorithms and structural graph theory.

Abstract

We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance grow arbitrarily. Some of these classes provide counter-examples to several conjectures. In particular, the construction includes hereditary classes of graphs whose treewidth is bounded by a function of the clique number while the tree-independence number is unbounded, thus disproving a conjecture of Dallard, Milanič and Štorgel [Treewidth versus clique number. II. Tree-independence number. Journal of Combinatorial Theory, Series B, 164:404-442, 2024.]. The construction can be further adjusted to provide, for any fixed integer $c$, graphs of arbitrarily large treewidth that contain no $K_c$-free graphs of high treewidth, thus disproving a conjecture of Hajebi [Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth, arXiv:2401.01299, 2024].

On treewidth and maximum cliques

TL;DR

The paper investigates the relationship between treewidth and clique number in hereditary graph classes by introducing -layered wheels, a flexible infinite construction that enforces holes of length at least while controlling clique growth across layers via a slow function . It proves a general width bound: every finite induced subgraph has bounded by a function of , yet the dependence can be tuned to be arbitrarily large, enabling a range of counterexamples. The authors then leverage this to disprove conjectures about tree-independence numbers and to challenge Hajebi’s proposals on -free obstructions, by producing graphs with large treewidth whose induced subgraphs avoid large-width configurations. The results establish a versatile toolkit for engineering width properties in hereditary classes and illuminate when width notions align or diverge, with implications for algorithms and structural graph theory.

Abstract

We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance grow arbitrarily. Some of these classes provide counter-examples to several conjectures. In particular, the construction includes hereditary classes of graphs whose treewidth is bounded by a function of the clique number while the tree-independence number is unbounded, thus disproving a conjecture of Dallard, Milanič and Štorgel [Treewidth versus clique number. II. Tree-independence number. Journal of Combinatorial Theory, Series B, 164:404-442, 2024.]. The construction can be further adjusted to provide, for any fixed integer , graphs of arbitrarily large treewidth that contain no -free graphs of high treewidth, thus disproving a conjecture of Hajebi [Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth, arXiv:2401.01299, 2024].
Paper Structure (5 sections, 20 theorems, 19 equations, 4 figures)

This paper contains 5 sections, 20 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of layered wheels
  3. Structure of layered wheels
  4. Finite induced subgraphs of layered wheels
  5. Applications of layered wheels

Key Result

Lemma 1.9

If $\mathcal{T} = (T, (X_s)_{s\in V (T )})$ is a tree decomposition of some graph $G$ and $(L_1, \dots, L_t)$ is a clique minor of $G$, then there exists $s\in V(T)$ such that $X_s$ contains at least one vertex of each $L_i$, $i\in \{1, \dots, t\}$. In particular, $\mathop{\mathrm{tw}}\nolimits(G) \

Figures (4)

  • Figure 1: Rules \ref{['a:hole']}--\ref{['aN']}
  • Figure 2: Rule \ref{['bS']}
  • Figure 3: Rule \ref{['bB']}
  • Figure 4: Layers $L_1$ to $L_4$ of the $(f, 4)$-layered wheel when $f(4) = 3$

Theorems & Definitions (41)

  • Conjecture 1.2: Dallard, Milanič and Štorgel, see DMS_JCTB2024
  • Conjecture 1.3: Hajebi, see hajebi:conj
  • Conjecture 1.4: Hajebi, see hajebi:conj
  • Conjecture 1.7: Hajebi, see hajebi:conj
  • Lemma 1.9
  • Theorem 1.10: Dvorák and Norin, see DBLP:journals/jct/DvorakN19
  • Theorem 1.11: Dirac, see dirac:chordal
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 31 more