Sebastian Wiederrecht, Delineating half-integrality of the Erdős-Pósa property for minors

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 minor 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.

In this talk, we start the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph $H$ there exists a finite family ${\mathfrak{F}}_H$ of parametric graphs such that $H$ has the Erdős-Pósa property in a minor-closed graph class $\mathcal{G}$ if and only if $\mathcal{G}$ excludes a minor of each of the parametric graphs in ${\mathfrak{F}}_H$. 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 family ${\mathfrak{F}}_H$ can be chosen to be precisely the two families of Robertson-Seymour counterexamples to the Erdős-Pósa property of $H$.