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Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors

Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

TL;DR

This work characterizes unavoidable induced subgraphs in graphs of large treewidth via two complementary frameworks. It proves that a graph with a sufficiently large complete bipartite induced minor $K_{f,f}$ must contain either a wall minor $W_{r\times r}$ or a large $(s,l)$-constellation, and then classifies the unavoidable structure inside large constellations into either interrupted or $2t$-zigzagged forms (with precise, finite parameters). The authors introduce and analyze constellations, mixed constellations, and the zigzagged framework, and they prove that both interrupted and zigzagged outcomes are necessary and cannot be replaced by other obstructions. The results connect induced minors, treewidth-driven obstructions, and highly structured subgraphs, establishing a foundation for a broader program in the series. These findings provide key ingredients for forthcoming work on the interplay between induced subgraphs and treewidth, with implications for understanding the density and structure of graphs with large dense minors.

Abstract

We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors

TL;DR

This work characterizes unavoidable induced subgraphs in graphs of large treewidth via two complementary frameworks. It proves that a graph with a sufficiently large complete bipartite induced minor must contain either a wall minor or a large -constellation, and then classifies the unavoidable structure inside large constellations into either interrupted or -zigzagged forms (with precise, finite parameters). The authors introduce and analyze constellations, mixed constellations, and the zigzagged framework, and they prove that both interrupted and zigzagged outcomes are necessary and cannot be replaced by other obstructions. The results connect induced minors, treewidth-driven obstructions, and highly structured subgraphs, establishing a foundation for a broader program in the series. These findings provide key ingredients for forthcoming work on the interplay between induced subgraphs and treewidth, with implications for understanding the density and structure of graphs with large dense minors.

Abstract

We prove that for every graph with a sufficiently large complete bipartite induced minor, either has an induced minor isomorphic to a large wall, or contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

Paper Structure

This paper contains 12 sections, 25 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Unavoidable constellations and the main result
  3. The zigzag graph
  4. Amplification
  5. Mixed constellations
  6. From mixed to zigzagged
  7. Finishing the proof of Theorem \ref{['thm:mainconstellation']}
  8. Complete bipartite induced minors
  9. The interrupted and the zigzagged outcome are both necessary
  10. The interrupted case
  11. The zigzagged case
  12. Acknowledgements

Key Result

Theorem 1.2

For every integer $r\in \mathbb{N}$, there is a constant $f_{thm:wallminor}=f_{thm:wallminor}(r)\in \mathbb{N}$ such that every graph $G$ with $\mathop{\mathrm{tw}}\nolimits(G) \geq f_{thm:wallminor}$ has a subgraph isomorphic to a subdivision of $W_{r\times r}$.

Figures (7)

  • Figure 1: The basic obstructions of treewidth 4: $K_{4,4}$ (top left), $K_5$ (bottom left), a subdivision of the $4 \times 4$ wall (middle) and the line graph of a subdivision of the $4 \times 4$ wall (right).
  • Figure 2: A Pohoata-Davies graph (left) and an occultation (right), both of treewidth $4$.
  • Figure 3: A $(3,3)$-constellation.
  • Figure 4: A $\mathfrak{c}$-route of length four in the $(3,3)$-constellation $\mathfrak{c}$ from Figure \ref{['fig:constellation']}, which is $2$-ample.
  • Figure 5: A $(4,1)$-constellation which is interupted with $x_1\preccurlyeq x_2\preccurlyeq x_3\preccurlyeq x_4$.
  • ...and 2 more figures

Theorems & Definitions (40)

  • Theorem 1.2: Robertson and Seymour GMV
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2: Aboulker, Adler, Kim, Sintiari, Trotignon; see Lemma 3.6 in aboulker
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2: Alecu, Chudnosvky, Hajebi and Spirkl; see Lemma 6.2 in tw9
  • Lemma 4.3: Hajebi pinned; see also Dvořák dvorak
  • ...and 30 more