the expansions of any given planar graph $J$ have bounding function $O (k \log k)$, where the hidden constant depends on the size of $J$. This is best possible when $J$ has a cycle (if $J$ is a forest then a $O(k)$ bound holds [@FJW2013]). Note that the Erdős-Pósa theorem can be viewed as a special case of this result, since each cycle is a $C_3$-expansion and every $C_3$-expansion contains a cycle as a subgraph.

While the vertex-EP property of expansions is now quite well understood, it was only recently realized (cp. [@RST16]) that the edge-EP property has not been thoroughly investigated. Since then, a new wave of papers has demonstrated the edge-EP property for $\theta_r$-expansions [@CRST18] (where $\theta_r$ is the multigraph consisting of two vertices joined by $r$ parallel edges), for *long cycles* [@BHJ19] (i.e.: the cycles of length at least some given constant $\ell$) and for $K_4$-expansions [@BH18]. There has also been much activity related to the edge-EP property for labeled graphs [@HU19; @B19] and directed graphs [@HU18]. From this account, one might gain the impression that the vertex-EP property and edge-EP property are essentially the same, and indeed sometimes the edge-EP property can be derived directly from 