paper is organised as follows. First, a sketch of the general approach is given in Section \[sec:sketch\]. Then we use it in Section \[sec:classic\_EP\] to give an alternative proof of the Erdős-Pósa theorem, and in Section \[sec:long\_cycles\] to prove Theorem \[thm:longcycles\]. Finally, we conclude the paper with some open problems (Section \[sec:open\_problems\]).

A sketch of the proof and beyond {#sec:sketch}
================================

We now provide a sketch of the general proof scheme used for Theorems \[thm:longcycles\] and \[thm:vertexcycle\], and how this might be useful to prove the edge-EP property for other graph classes as well.\
Suppose we are given a fixed graph $J$ and we want to prove the edge-EP property for the class of $J-$expansions, with an $O(k \log k)$ bounding function. Let $k \in \mathbb{N}$ and let $G$ be a graph. The first step is to apply induction on the number of edges of $G$. If $G$ has a subgraph $H$ which is a small $J$-expansion, of size at most $O (\log k)$, then one can apply a straightforward induction to the graph $G-E(H)$. Hence we may assume that each $J-$expansion in $G$ is large, of size $\Omega(\log k)$. Similarly, if we can apply a reduction operation on $G$, 