Delineating Half-Integrality of the Erd\H{o}s-P\'osa Property for Minors: the Case of Surfaces

• Main Authors: Paul, Christophe, Protopapas, Evangelos, Thilikos, Dimitrios M, Wiederrecht, Sebastian
• Format: Journal Article
• Language: English
• Published: 24-06-2024
• Subjects: Computer Science - Discrete Mathematics; Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2406.16647

In 1986 Robertson and Seymour proved a generalization of the seminal result
of Erd\H{o}s and P\'osa on the duality of packing and covering cycles: A graph
has the Erd\H{o}s-P\'osa property for minors if and only if it is planar. In
particular, for every non-planar graph $H$ they gave examples showing that the
Erd\H{o}s-P\'osa property does not hold for $H.$ Recently, Liu confirmed a
conjecture of Thomas and showed that every graph has the half-integral
Erd\H{o}s-P\'osa 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
Erd\H{o}s-P\'osa 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 Erd\H{o}s-P\'osa 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
Erd\H{o}s-P\'osa 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 $\mathcal{H}.$