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Excluding a Forest Induced Minor

Édouard Bonnet, Benjamin Duhamel, Robert Hickingbotham

TL;DR

This work completes a dichotomy for forests H in the induced-minor setting: in weakly sparse graph classes, excluding $H$ as an induced minor yields bounded pathwidth precisely when $H$ belongs to a specific forest family $\mathcal{F}$ built as the induced-minor closure of two infinite constructions. The authors leverage the constellation framework (interrupted and $q$-zigzagged constellations) to connect large pathwidth to unavoidable induced substructures, while canonical obstructions (Pohoata–Davies grids and death stars) prove negative cases. They classify forests into negative and positive, showing forests not induced-minors of $T_{2,\ell}\cup \ell T_{1,\ell}$ or $T_{3,\ell}\cup \ell T_{1,\ell}$ are negative and those that are are positive, thereby deriving a complete characterization and a linear-time detection consequence in weakly sparse classes. The results advance the broader goal of classifying all $H$ for which every weakly sparse $H$-induced-minor-free class has bounded treewidth or pathwidth, and connect to induced-subgraph decomposition techniques via constellations.

Abstract

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been devoted to understanding the unavoidable induced substructures of graphs with large pathwidth or large treewidth. In this paper, we give an induced counterpart of the Forest Minor theorem: for any $t \geqslant 2$, the $K_{t,t}$-subgraph-free $H$-induced-minor-free graphs have bounded pathwidth if and only if $H$ belongs to a class $\mathcal F$ of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs $H$ for which every weakly sparse $H$-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.

Excluding a Forest Induced Minor

TL;DR

This work completes a dichotomy for forests H in the induced-minor setting: in weakly sparse graph classes, excluding as an induced minor yields bounded pathwidth precisely when belongs to a specific forest family built as the induced-minor closure of two infinite constructions. The authors leverage the constellation framework (interrupted and -zigzagged constellations) to connect large pathwidth to unavoidable induced substructures, while canonical obstructions (Pohoata–Davies grids and death stars) prove negative cases. They classify forests into negative and positive, showing forests not induced-minors of or are negative and those that are are positive, thereby deriving a complete characterization and a linear-time detection consequence in weakly sparse classes. The results advance the broader goal of classifying all for which every weakly sparse -induced-minor-free class has bounded treewidth or pathwidth, and connect to induced-subgraph decomposition techniques via constellations.

Abstract

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the -minor-free graphs have bounded pathwidth if and only if is a forest. In recent years, considerable effort has been devoted to understanding the unavoidable induced substructures of graphs with large pathwidth or large treewidth. In this paper, we give an induced counterpart of the Forest Minor theorem: for any , the -subgraph-free -induced-minor-free graphs have bounded pathwidth if and only if belongs to a class of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs for which every weakly sparse -induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.

Paper Structure

This paper contains 8 sections, 16 theorems, 8 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Proof Outline
  3. Preliminaries
  4. Standard graph-theoretic definitions and notation
  5. Constellations
  6. The Pohoata--Davies grid and the Death Star
  7. Negative Forests
  8. Positive Forests

Key Result

Theorem 1

For every graph $F$ and integer $t \geqslant 2$, the class of graphs excluding $F$ as an induced minor and $K_{t,t}$ as a subgraph has bounded pathwidth if and only if $F$ is a forest without either of the following:

Figures (13)

  • Figure 1: The $6 \times 6$ Pohoata--Davies grid.
  • Figure 2: Two minimal trees satisfying \ref{['lem:npd']}, thus not being induced minors of any Pohoata--Davies grid.
  • Figure 3: The death star of height 5.
  • Figure 4: Two minimal trees satisfying \ref{['lem:nds']}, thus not being induced minors of any death star. Left: no edge between $A \cup B$ and $C \cup D$. Right: one permitted (unique) edge between $A \cup B$ and $C \cup D$, namely between $B$ and $C$. It is indeed incident to two claw leaves outside $V(P) \cup V(Q)$.
  • Figure 5: $T_{1,\ell}$ for $\ell=5$ (left) and its shorthand representation (right).
  • ...and 8 more figures

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2: reformulation of \ref{['thm:main']}
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5: Theorem 2.1 in istd18
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 17 more