Condensed-wall criterion for the edge-Erdős–Pósa property

Let $H$ be a planar graph. An $H$-expansion is a subdivision of $H$, and a condensed wall is the graph class described in the paper. The edge-Erdős–Pósa property concerns the existence of bounded edge hitting sets for families of pairwise edge-disjoint subgraphs.

Condensed-wall conjecture. If there is an integer $r$ such that the condensed wall of size $r$ contains an $H$-expansion, then the family of $H$-expansions has the edge-Erdős–Pósa property.

The conjecture is motivated by the known positive cases for long cycles, theta-graphs, and $K_4$, each of which occurs as a minor in a sufficiently large condensed wall. The paper presents the condensed-wall containment condition as a proposed sufficient criterion, but does not prove it in general.