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Tree independence number V. Walls and claws

Maria Chudnovsky, Julien Codsi, Daniel Lokshtanov, Martin Milanič, Varun Sivashankar

TL;DR

This work addresses bounding the tree independence number for graphs that exclude the families $\mathcal{L}_t$, $S_{t,t,t}$, and $K_{t,t}$. The authors develop a structural framework built around extended strip decompositions and a layered-sets technique, establishing that the underlying pattern $H$ in such decompositions has bounded treewidth, which yields small cores for balanced separators. Through a sequence of boosting and disjointness arguments, they obtain $(w,\varepsilon)$-boosted separators with polylogarithmic core sizes, culminating in a global separator yielding a tree independence bound of $c(t)\log^4 n$. Consequently, Maximum Weight Independent Set and related problems become quasi-polynomial-time solvable on these graph classes. The paper also proves a polylogarithmic bound on tree independence using a $d(t)$-breakable property for the class $\mathcal{M}_t^*$, contributing to a broader program of understanding induced-subgraph obstructions to small tree independence numbers.

Abstract

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $S_{t,t,t}$ be the graph obtained from $K_{1,3}$ by subdividing each edge $t-1$ times, and let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid. Let $\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \times t}$. We prove that for every positive integer $t$ there exists $c(t)$ such that every $\mathcal{L}_t \cup \{S_{t,t,t}, K_{t,t}\}$-free $n$-vertex graph admits a tree decomposition in which the maximum size of an independent set in each bag is at most $c(t)\log^4n$. This is a variant of a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht from 2024. This implies that the Maximum Weight Independent Set problem, as well as many other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is $\mathcal{L}_t \cup \{S_{t,t,t},K_{t,t}\}$-free. As part of our proof, we show that for every positive integer $t$ there exists an integer $d$ such that every $\mathcal{L}_t \cup \{S_{t,t,t}\}$-free graph admits a balanced separator that is contained in the neighborhood of at most $d$ vertices.

Tree independence number V. Walls and claws

TL;DR

This work addresses bounding the tree independence number for graphs that exclude the families , , and . The authors develop a structural framework built around extended strip decompositions and a layered-sets technique, establishing that the underlying pattern in such decompositions has bounded treewidth, which yields small cores for balanced separators. Through a sequence of boosting and disjointness arguments, they obtain -boosted separators with polylogarithmic core sizes, culminating in a global separator yielding a tree independence bound of . Consequently, Maximum Weight Independent Set and related problems become quasi-polynomial-time solvable on these graph classes. The paper also proves a polylogarithmic bound on tree independence using a -breakable property for the class , contributing to a broader program of understanding induced-subgraph obstructions to small tree independence numbers.

Abstract

Given a family of graphs, we say that a graph is -free if no induced subgraph of is isomorphic to a member of . Let be the graph obtained from by subdividing each edge times, and let be the -by- hexagonal grid. Let be the family of all graphs such that is the line graph of some subdivision of . We prove that for every positive integer there exists such that every -free -vertex graph admits a tree decomposition in which the maximum size of an independent set in each bag is at most . This is a variant of a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht from 2024. This implies that the Maximum Weight Independent Set problem, as well as many other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is -free. As part of our proof, we show that for every positive integer there exists an integer such that every -free graph admits a balanced separator that is contained in the neighborhood of at most vertices.
Paper Structure (12 sections, 27 theorems, 33 equations, 11 figures, 1 algorithm)

This paper contains 12 sections, 27 theorems, 33 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proof outline and organization
  3. Constricted sets and extended strip decompositions
  4. Extended strip decompositions in graphs in $\mathcal{M}_t^*$
  5. Bounded core separators in graphs in $\mathcal{M}_t^*$
  6. Large stable sets in neighborhoods
  7. The layered sets argument
  8. Improving the separators
  9. Improving the Separating Power
  10. Improving the Disjointedness of the Separators
  11. Bringing it all together
  12. Acknowledgment

Key Result

Theorem 1.2

For every positive integer $t$ there exists $c=c(t)$ such that every $n$-vertex graph in $\mathcal{M}_t$ with $n\ge 2$ has tree independence number at most $c \log^4 n$.

Figures (11)

  • Figure 1: Example of an extended strip decomposition with its pattern (here dash lines represent potential edges). This figure was created by Paweł Rzążewski and we use it with his permission.
  • Figure 2: Example of $H'$ given an $H$.
  • Figure 3: Visualization for \ref{['noskip']}.
  • Figure 4: Visualization of $h_0$ and $h_1$.
  • Figure 5: Visualization of $G_h$ (here dashed lines represent non-edges).
  • ...and 6 more figures

Theorems & Definitions (48)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • ...and 38 more