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Delineating Half-Integrality of the Erdős-Pósa Property for Minors: the Case of Surfaces

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos, Sebastian Wiederrecht

Abstract

In 1986 Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minors if and only if it is planar. In particular, for every non-planar graph $H$ they gave examples showing that the Erdős-Pósa property does not hold for $H.$ Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. Liu's proof is non-constructive and to this date, with the exception of a small number of examples, no constructive proof is known. In this paper, we initiate the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph $H,$ there exists a unique (up to a suitable equivalence relation) graph parameter ${\textsf{EP}}_H$ such that $H$ has the Erdős-Pósa property in a minor-closed graph class $\mathcal{G}$ if and only if $\sup\{\textsf{EP}_H(G) \mid G\in\mathcal{G}\}$ is finite. We prove this conjecture for the class $\mathcal{H}$ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar $H\in\mathcal{H},$ the parameter ${\sf EP}_H(G)$ is precisely the maximum order of a Robertson-Seymour counterexample to the Erdős-Pósa property of $H$ which can be found as a minor in $G.$ Our results are constructive and imply, for the first time, parameterized algorithms that find either a packing, or a cover, or one of the Robertson-Seymour counterexamples, certifying the existence of a half-integral packing for the graphs in $\mathcal{H}.$

Delineating Half-Integrality of the Erdős-Pósa Property for Minors: the Case of Surfaces

Abstract

In 1986 Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minors if and only if it is planar. In particular, for every non-planar graph they gave examples showing that the Erdős-Pósa property does not hold for Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. Liu's proof is non-constructive and to this date, with the exception of a small number of examples, no constructive proof is known. In this paper, we initiate the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph there exists a unique (up to a suitable equivalence relation) graph parameter such that has the Erdős-Pósa property in a minor-closed graph class if and only if is finite. We prove this conjecture for the class of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar the parameter is precisely the maximum order of a Robertson-Seymour counterexample to the Erdős-Pósa property of which can be found as a minor in Our results are constructive and imply, for the first time, parameterized algorithms that find either a packing, or a cover, or one of the Robertson-Seymour counterexamples, certifying the existence of a half-integral packing for the graphs in
Paper Structure (101 sections, 55 theorems, 62 equations, 6 figures)

This paper contains 101 sections, 55 theorems, 62 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Half-integral Erdős-Pósa.
  3. The threshold of half-integrality
  4. Kuratowski-connectivity.
  5. Shallow-vortex minors.
  6. Our contribution.
  7. Notation and definitions
  8. Equivalence of graph parameters.
  9. Surfaces and embeddability.
  10. Parametric graphs and Dyck-grids.
  11. Our results
  12. Some implications of our results
  13. Half-integral Erdős-Pósa for linked pairs and knots.
  14. Outline of the proof
  15. Tree decompositions.
  16. ...and 86 more sections

Key Result

Theorem 1.1

For every $\mathcal{Z}\in\mathbb{H}^{-},$ for every minor-closed graph class $\mathcal{G}$, $\mathcal{Z}$ has the Erdős-Pósa property in $\mathcal{G}$ if and only if $\text{\scriptsizeEP}_{\mathcal{Z}}$ is bounded in $\mathcal{G}$.

Figures (6)

  • Figure 1: The parametric graphs representing the annulus grid $\mathscr{A}_{k},$ the handle grid $\mathscr{H}_{k},$ the cross-cap grid $\mathscr{C}_{k},$ and the shallow-vortex grid $\mathscr{V}_{k}.$
  • Figure 2: The two first parametric graphs serve as counterexamples for the Erdős-Pósa property of the graph $J$. The third parametric graph is a counterexample for the Erdős-Pósa property of $K_8.$ All three parametric graphs have unbounded Euler-genus. For the first two this is witnessed by a large packing of $K_{3,3}$ while the last one can be observed to contain $K_{3,r}$ as a minor.
  • Figure 3: The Dyck-grid $\mathscr{D}_{8}^{1,2}$. The simple and the exceptional cycles are drawn in orange.
  • Figure 4: The elementary $(h,c;k)$-Dyck wall, where $h=1,$$c=1,$ and $k=6.$ We draw in magenta one of the tracks of the $(h,c;k)$-Dyck wall.
  • Figure 5: The (partial) drawing of the shallow-vortex grid $\mathscr{V}_{8}$ where 20 or its 32 cycles are drawn along with the drawing of a subdivision of $\mathscr{V}_{2}$ in it. Notice that $\mathscr{V}_{2}$ is a minor of $\mathscr{V}_{6}.$ In general, $\mathscr{V}_{k}$ can be seen as a minor of $\mathscr{V}_{3k},$ using (parts of) $6\times 4$ tracks, because we need paths from $k$ more tracks on the left along with paths from $k$ tracks on the right, and paths from $k$ more cycles below (all drawn in violet) in order to simulate the $k$ missing edges completing the $k$ cycles of $\mathscr{V}_{2}.$
  • ...and 1 more figures

Theorems & Definitions (95)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1: thilikos2023excluding
  • Proposition 2
  • Proposition 3: kawarabayashi2020quickly
  • Proposition 4: gavoille2023minoruniversal
  • Proposition 5: thilikos2023excluding
  • ...and 85 more