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

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 $\mathcal{H}.$