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A grid theorem for strong immersions of walls

Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Paul Wollan

TL;DR

The paper addresses when a graph contains a large wall $W_\ell$ as a strong immersion minor and establishes a grid-theorem-like dichotomy: either such a wall is present or the graph admits a tree-cut decomposition with bounded adhesion whose torsos are governed by almost $\alpha$-thin $3$-centres. The authors develop the notions of almost $\alpha$-thin graphs and $3$-centres, reduce to the $3$-edge-connected case, and prove a key obstruction showing that large walls cannot occur under these decompositions. They then provide a constructive proof that, given $\ell$, one can choose $\alpha(\ell)$ so that the absence of a strong immersion of $W_\ell$ implies the desired decomposition, while conversely, a decomposition with almost $\alpha$-thin $3$-centres forbids large walls as strong immersions. The results yield a structural understanding of strong immersions of walls akin to the classical grid theorem for minors, with potential algorithmic implications for immersion-robust graph classes, and emphasize the essential role of $3$-centres in this setting.

Abstract

We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small `width', which is measured in terms of its adhesion and the path-likeness of its torsos.

A grid theorem for strong immersions of walls

TL;DR

The paper addresses when a graph contains a large wall as a strong immersion minor and establishes a grid-theorem-like dichotomy: either such a wall is present or the graph admits a tree-cut decomposition with bounded adhesion whose torsos are governed by almost -thin -centres. The authors develop the notions of almost -thin graphs and -centres, reduce to the -edge-connected case, and prove a key obstruction showing that large walls cannot occur under these decompositions. They then provide a constructive proof that, given , one can choose so that the absence of a strong immersion of implies the desired decomposition, while conversely, a decomposition with almost -thin -centres forbids large walls as strong immersions. The results yield a structural understanding of strong immersions of walls akin to the classical grid theorem for minors, with potential algorithmic implications for immersion-robust graph classes, and emphasize the essential role of -centres in this setting.

Abstract

We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small `width', which is measured in terms of its adhesion and the path-likeness of its torsos.
Paper Structure (7 sections, 14 theorems, 4 equations, 1 figure)

This paper contains 7 sections, 14 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Almost alpha-thin graphs and 3-centres
  3. Strongly immersed walls are obstructions
  4. Excluding strongly immersed walls
  5. Reduction to 3-edge-connected graphs
  6. Immersions and tree-cut de-com-po-si-tion s
  7. Proof of Theorem 1

Key Result

Theorem 1.1

For every graph $F$, there exists an integer $\alpha = \alpha(F) > 0$ such that if a graph $G$ does not contain $F$ as a strong immersion minor, then there exists a tree-cut de-com-po-si-tion of $G$ of adhesion less than $\alpha$ such that each of its torsos is $\alpha$-basic.

Figures (1)

  • Figure 1: The wall $W_4$ of size $4$ with the underlying $4 \times 8$ grid.

Theorems & Definitions (33)

  • Theorem 1.1: dvorak2016structure*Theorem 4
  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Example 2.2
  • proof
  • Example 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 23 more