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Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star

Clément Dallard, Matjaž Krnc, O-joung Kwon, Martin Milanič, Andrea Munaro, Kenny Štorgel, Sebastian Wiederrecht

TL;DR

This paper studies when bounded treewidth in hereditary graph classes is governed by large cliques, via the tree-independence number $\mathsf{tree}\text{-}\alpha(G)$. It proves that for graph classes defined by finitely many forbidden induced subgraphs containing an induced star, $(tw,\omega)$-boundedness is equivalent to bounded $\mathsf{tree}\text{-}\alpha$, and extends this to sub-classes of line graphs. It also determines exact $\mathsf{tree}\text{-}\alpha$ values for line graphs of complete graphs and complete bipartite graphs, and gives a linear-time algorithm for computing $\mathsf{tree}\text{-}\alpha$ in $P_4$-free graphs, with polynomial-time MWIS algorithms for broad families. The results bridge structural graph theory with algorithmic consequences, offering concrete bounds and constructive procedures that apply to several important graph classes and problems. This advances understanding of how induced-obstruction constraints influence tractable independent-set computations and related width parameters.

Abstract

Many recent works address the question of characterizing induced obstructions to bounded treewidth. In 2022, Lozin and Razgon completely answered this question for graph classes defined by finitely many forbidden induced subgraphs. Their result also implies a characterization of graph classes defined by finitely many forbidden induced subgraphs that are $(tw,ω)$-bounded, that is, treewidth can only be large due to the presence of a large clique. This condition is known to be satisfied for any graph class with bounded tree-independence number, a graph parameter introduced independently by Yolov in 2018 and by Dallard, Milanič, and Štorgel in 2024. Dallard et al. conjectured that $(tw,ω)$-boundedness is actually equivalent to bounded tree-independence number. We address this conjecture in the context of graph classes defined by finitely many forbidden induced subgraphs and prove it for the case of graph classes excluding an induced star. We also prove it for subclasses of the class of line graphs, determine the exact values of the tree-independence numbers of line graphs of complete graphs and line graphs of complete bipartite graphs, and characterize the tree-independence number of $P_4$-free graphs, which implies a linear-time algorithm for its computation. Applying the algorithmic framework provided in a previous paper of the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes.

Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star

TL;DR

This paper studies when bounded treewidth in hereditary graph classes is governed by large cliques, via the tree-independence number . It proves that for graph classes defined by finitely many forbidden induced subgraphs containing an induced star, -boundedness is equivalent to bounded , and extends this to sub-classes of line graphs. It also determines exact values for line graphs of complete graphs and complete bipartite graphs, and gives a linear-time algorithm for computing in -free graphs, with polynomial-time MWIS algorithms for broad families. The results bridge structural graph theory with algorithmic consequences, offering concrete bounds and constructive procedures that apply to several important graph classes and problems. This advances understanding of how induced-obstruction constraints influence tractable independent-set computations and related width parameters.

Abstract

Many recent works address the question of characterizing induced obstructions to bounded treewidth. In 2022, Lozin and Razgon completely answered this question for graph classes defined by finitely many forbidden induced subgraphs. Their result also implies a characterization of graph classes defined by finitely many forbidden induced subgraphs that are -bounded, that is, treewidth can only be large due to the presence of a large clique. This condition is known to be satisfied for any graph class with bounded tree-independence number, a graph parameter introduced independently by Yolov in 2018 and by Dallard, Milanič, and Štorgel in 2024. Dallard et al. conjectured that -boundedness is actually equivalent to bounded tree-independence number. We address this conjecture in the context of graph classes defined by finitely many forbidden induced subgraphs and prove it for the case of graph classes excluding an induced star. We also prove it for subclasses of the class of line graphs, determine the exact values of the tree-independence numbers of line graphs of complete graphs and line graphs of complete bipartite graphs, and characterize the tree-independence number of -free graphs, which implies a linear-time algorithm for its computation. Applying the algorithmic framework provided in a previous paper of the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes.
Paper Structure (17 sections, 37 theorems, 18 equations, 3 figures)

This paper contains 17 sections, 37 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Our focus
  4. Our results
  5. Algorithmic implications
  6. Structure of the paper
  7. Preliminaries
  8. Tree-independence number of {K_{1,d},Pₛ}-free graphs
  9. Tree-independence number of {K_{1,d},S,T}-free graphs
  10. Case 1: $w_1$ has a neighbor in $s_{r+p+1}P^1s_{m-r-p}$.
  11. Case 2: All neighbors of $w_1$ on $P^1$ belong either to $s_1P^1s_{r+p}$ or to $s_{m-r-p+1}P^1s_m$.
  12. Tree-independence number of line graphs
  13. Conjecture 1.1 for subclasses of the class of line graphs
  14. Tree-independence number of L(K_{n,n}) and L(K_{n})
  15. Tree-independence number of P₄-free graphs
  16. ...and 2 more sections

Key Result

Theorem 1.4

For any positive integer $d$ and any two graphs $S\in \mathcal{S}$ and $T\in L(\mathcal{S})$, the class of $\{K_{1,d},S,T\}$-free graphs has bounded tree-independence number.

Figures (3)

  • Figure 1: The paths $P^1$, $F$, and $Q$ in \ref{['thm:excludelongclaw']}.
  • Figure 2: An illustration for the proof of \ref{['thm:line']}: (a) shows the graph $G_3$ together with the edge coloring, (b) is the line graph $H_3$ of $G_3$ with the vertex coloring induced by the edge coloring of $G_3$, and (c) shows the resulting $K_{3,3}$ after contracting the dashed edges in (b).
  • Figure 3: The elementary $4$-wall.

Theorems & Definitions (62)

  • Conjecture 1.1: Dallard et al. dallard2022secondpaper
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 2.1: folklore
  • Theorem 2.2
  • ...and 52 more