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Quickly excluding an annotated planar graph

Maximilian Gorsky, Evangelos Protopapas, Sebastian Wiederrecht

TL;DR

This work delivers a constructive, polynomial-bounded proof of the structure theorem for annotated graphs with bounded bidimensionality, extending the Grid Theorem paradigm to red minors (rooted/minor-annotated frameworks). The authors develop a comprehensive toolkit—flat walls, society classifications, and local-to-global decompositions—ensuring all parameters in the theorem are bounded by a polynomial in k, and provide an algorithm that either certifies bidimensionality ≥ k or yields a near-embedding with polynomially bounded complexity. By combining advances in polynomial bounds within Graph Minor theory with annotated-vertex-set techniques, they also sketch polynomial bounds for apex-minor excluded graphs, and discuss implications for extensions of Courcelle’s Theorem and Steiner Tree-type problems on annotated graphs. The results illuminate how restricting the red-vertex structure relative to a minor-free graph class yields tractable, structure-driven algorithmic consequences, highlighting the central role of bidimensionality in contemporary parameterized and structural graph theory. Overall, the paper advances a constructive, polynomially-bounded framework for annotated graphs that unifies several strands of minor theory, algorithmic meta-theorems, and layered decompositions. The methods promise broad applicability to problems like Steiner Tree in apex- and apex-minor-free classes, and to the broader program of extending Courcelle-type results under annotated and near-embedding constraints.

Abstract

We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle's Theorem to $H$-minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set $R$. A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever $R$ can be covered with few faces. Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph $G$ can be decomposed into pieces where the behaviour of $R$ is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour's celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.

Quickly excluding an annotated planar graph

TL;DR

This work delivers a constructive, polynomial-bounded proof of the structure theorem for annotated graphs with bounded bidimensionality, extending the Grid Theorem paradigm to red minors (rooted/minor-annotated frameworks). The authors develop a comprehensive toolkit—flat walls, society classifications, and local-to-global decompositions—ensuring all parameters in the theorem are bounded by a polynomial in k, and provide an algorithm that either certifies bidimensionality ≥ k or yields a near-embedding with polynomially bounded complexity. By combining advances in polynomial bounds within Graph Minor theory with annotated-vertex-set techniques, they also sketch polynomial bounds for apex-minor excluded graphs, and discuss implications for extensions of Courcelle’s Theorem and Steiner Tree-type problems on annotated graphs. The results illuminate how restricting the red-vertex structure relative to a minor-free graph class yields tractable, structure-driven algorithmic consequences, highlighting the central role of bidimensionality in contemporary parameterized and structural graph theory. Overall, the paper advances a constructive, polynomially-bounded framework for annotated graphs that unifies several strands of minor theory, algorithmic meta-theorems, and layered decompositions. The methods promise broad applicability to problems like Steiner Tree in apex- and apex-minor-free classes, and to the broader program of extending Courcelle-type results under annotated and near-embedding constraints.

Abstract

We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle's Theorem to -minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set . A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever can be covered with few faces. Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph can be decomposed into pieces where the behaviour of is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour's celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.
Paper Structure (74 sections, 37 theorems, 46 equations, 9 figures)

This paper contains 74 sections, 37 theorems, 46 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Our result
  3. Related work, consequences, and applications
  4. Generalising treewidth through annotation.
  5. An application: Excluding an apex minor.
  6. The algorithmic potential of annotated graph parameters.
  7. Layered parameters and the exclusion of "apex-$\mathcal{X}$" graphs.
  8. Towards colorful minors of $q$-colorful graphs.
  9. Overview of our proof
  10. Step 1: A flat wall.
  11. Step 2: Classifying a red society.
  12. Step 3: The local structure theorem.
  13. Step 4: Local to global.
  14. Preliminaries
  15. Basics
  16. ...and 59 more sections

Key Result

Theorem 1.1

There exists a function $f_{thm:BidimensionalityIntro}\colon\mathbb{N}\to\mathbb{N}$ such that for all non-negative integers $k$, and all annotated graphs $(G,R)$ one of the following holds: Moreover, $f_{thm:BidimensionalityIntro}(k)\in k^{\mathbf{O}(1)}$ and there exists an algorithm that finds either a $(k\times k)$-grid-minor-model witnessing that $(G,R)$ has bidimensionality at least $k$, or

Figures (9)

  • Figure 1: Diagrams of (i) the red $(7 \times 7)$-grid and two examples illustrating that the structure emerging from excluding a red grid resembles the structure of excluding an arbitrary minor. (ii) A grid with its outermost column being red: no large red grid exists but at the same time the red vertices cannot be easily removed from the grid. (iii) a more complicated example where embeddability can only be achieved by deleting an apex vertex and few faces do not suffice to describe the structure of the red vertices.
  • Figure 2: A model of a red $(5 \times 5)$-grid.
  • Figure 3: Diagrams of (i) a division of a $(5+4r)$-wall to either find a an $r$-subwall without any red vertices, or a red $(4\times 4)$-grid minor and (ii) a $(4 \times 4)$-grid minor in the second outcome.
  • Figure 4: Diagrams of (i) a crosscap transaction, (ii) a handle transaction, and (iii) a weak near embedding in a disk with a small number of vortices.
  • Figure 5: Diagrams of (i) a strip of a transaction divided into $16$ pairwise disjoint substrips, each containing a transaction of order $r$ and (ii) a $(4 \times 4)$-grid minor in the case where all strips in (i) contain a red vertex.
  • ...and 4 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Protopapas, Thilikos, and Wiederrecht ProtopapasTW2025Colorful
  • Proposition 3.1: Gorsky, Seweryn, and Wiederrecht GorskySW2025Polynomial
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 47 more