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

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

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Author Details

Christophe Paul
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Evangelos Protopapas
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Dimitrios M. Thilikos
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Sebastian Wiederrecht
  • Discrete Mathematics Group, Institute for Basic Science, Daejeon, South Korea

Funding

  • Paul, Christophe: Supported by the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).
  • Protopapas, Evangelos: Supported by the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).
  • Thilikos, Dimitrios M.: Supported by the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027) and the MEAE and the MESR via the Franco-Norwegian project PHC Aurora project n. 51260WL (2024).
  • Wiederrecht, Sebastian: Supported by the Institute for Basic Science (IBS-R029-C1).

Cite As Get BibTex

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos, and Sebastian Wiederrecht. Delineating Half-Integrality of the Erdős-Pósa Property for Minors: The Case of Surfaces. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 114:1-114:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.114

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 on graph parameters) graph parameter EP_H such that H has the Erdős-Pósa property in a minor-closed graph class 𝒢 if and only if sup{EP_H(G) ∣ G ∈ 𝒢} is finite. We prove this conjecture for the class ℋ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar H ∈ ℋ, the parameter 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 ℋ.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
Keywords
  • Erdős-Pósa property
  • Erdős-Pósa pair
  • Graph parameters
  • Graph minors
  • Universal obstruction
  • Surface containment

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