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Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

Maria Chudnovsky, Julien Codsi, J. Pascal Gollin, Martin Milanič, Varun Sivashankar

Abstract

We show that for every positive integer ${t \geq 2}$ there exists an integer $s$ such that every graph that contains no induced subgraph isomorphic to either the $6$-vertex path or the $(2,t)$-biclique, the complete bipartite graph $K_{2,t}$, has tree-independence number at most $s$. This result makes partial progress on a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.

Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

Abstract

We show that for every positive integer there exists an integer such that every graph that contains no induced subgraph isomorphic to either the -vertex path or the -biclique, the complete bipartite graph , has tree-independence number at most . This result makes partial progress on a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.

Paper Structure

This paper contains 14 sections, 15 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper and overview of our techniques
  3. Preliminaries
  4. Basic notions and notation
  5. Paths and separators
  6. The connectifier lemma
  7. Tree-decompositions, treewidth, and tree-independence number
  8. Vertex-weighted graphs and balanced separators
  9. The main theorem
  10. Pyramids in P_6 , K_2,t -free graphs
  11. Structure of pyramids in P_6 , K_2,t -free graphs
  12. Existence of pyramids in P_6 , K_2,t -free graphs
  13. Proof of Lemma 3.1
  14. Proof of Lemma 3.2

Key Result

Theorem 1.2

For every integer ${t \geq 2}$ there exists an integer $s$ such that every ${\{P_6,K_{2,t}\}}$-free graph has tree-independence number at most $s$.

Figures (2)

  • Figure 4.1: The situation in the proof of lem:pyramids-exist-2. The dotted edges may or may not be edges of $G$.
  • Figure 4.2: The six cases from the proof of lem:pyramids-exist-1, illustrated for the case $t = 3$. Top row: two stars ($K_{2,t}$), a star and a $1$-subdivision of a star (contains an induced $P_6$), and two $1$-subdivisions of a star (contains an induced $P_9$). Bottom row: the line graph of the $1$-subdivision of a star and the $1$-subdivision of a star (contains an induced $P_8$), two line graphs of $1$-subdivisions of stars (contains an induced $P_7$), and a line graph of the $1$-subdivision of a star and a star (the desired pyramid).

Theorems & Definitions (44)

  • Conjecture 1.1: Dallard et al. DallardKKMMSW:2024:TWvsOmega4
  • Conjecture 1.2
  • Theorem 1.2
  • Lemma 2.1: The connectifier lemma, Chudnovsky et al. ChudnovskyGHLS:2024:tw15
  • Lemma 2.3: Chudnovsky et al. ChudnovskiHLS2026:TIN2
  • Theorem 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.1
  • ...and 34 more