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Tree-independence number VII. Excluding a star

Maria Chudnovsky, Jadwiga Czyżewska, Marcin Pilipczuk, Paweł Rzążewski

TL;DR

The paper addresses bounding the tree-independence number for graphs that are both $H$-induced-minor-free with $H$ planar and $K_{1,s}$-free. It combines structural ingredients—embedding $H$ into a hexagonal grid $W_{r\times r}$, constellation configurations arising from $K_{t,t}$ minors, and $\lambda$-separability—to reduce to a bounded-treewidth setting, then uses Hajebi's polynomial-treewidth bound and known equivalences to derive a polylogarithmic bound on tree-independence: $\mathrm{tree\_alpha}(G) \le \log^{c_{s,H}} n$. This provides strong evidence that the dense-graph analogue of the conjecture holds up to polylogarithmic factors, marking progress toward the full resolution. The results illuminate how exclusion of induced minors and specific subgraphs constrains decompositions and highlights the interplay between treewidth, clique structure, and independence within tree decompositions.

Abstract

We prove that for every fixed integer $s$ and every planar graph $H$, the class of $H$-induced-minor-free and $K_{1,s}$-induced-subgraph-free graphs has polylogarithmic tree-independence number. This is a weakening of a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.

Tree-independence number VII. Excluding a star

TL;DR

The paper addresses bounding the tree-independence number for graphs that are both -induced-minor-free with planar and -free. It combines structural ingredients—embedding into a hexagonal grid , constellation configurations arising from minors, and -separability—to reduce to a bounded-treewidth setting, then uses Hajebi's polynomial-treewidth bound and known equivalences to derive a polylogarithmic bound on tree-independence: . This provides strong evidence that the dense-graph analogue of the conjecture holds up to polylogarithmic factors, marking progress toward the full resolution. The results illuminate how exclusion of induced minors and specific subgraphs constrains decompositions and highlights the interplay between treewidth, clique structure, and independence within tree decompositions.

Abstract

We prove that for every fixed integer and every planar graph , the class of -induced-minor-free and -induced-subgraph-free graphs has polylogarithmic tree-independence number. This is a weakening of a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.
Paper Structure (7 sections, 9 theorems, 1 equation)

This paper contains 7 sections, 9 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notation and tools
  3. Graphs.
  4. Tree decompositions.
  5. Building blocks
  6. Proof of \ref{['thm:main']}
  7. Conclusion

Key Result

Theorem 1.1

For every planar graph $H$ there exists an integer $c_H$ such that every graph that does not contain a minor isomorphic to $H$ has treewidth at most $c_H$.

Theorems & Definitions (12)

  • Theorem 1.1: Robertson, Seymour GMV
  • Theorem 1.2: Korhonen korhonen2023grid
  • Conjecture 1.3: DBLP:journals/corr/abs-2402-11222
  • Theorem 1.4
  • Theorem 2.1: Ramsey Ramsey, see also Erdős-Szekeres ErdosSzekeres:1935:ACombinatorialProblemInGeometry
  • Theorem 2.2
  • Theorem 2.3: Chudnovsky, Hajebi, Spirkl chudnovsky2025inducedsubgraphstreedecompositions
  • Theorem 2.4: Hajebi hajebi2025polynomialboundspathwidth
  • Theorem 2.5: Chudnovsky, Lokshtanov, Satheeshkumar chudnovsky2025treewidthcliqueboundednesspolylogarithmictreeindependence
  • Theorem 3.1
  • ...and 2 more