is edge-disjoint from $T_1$ and $T_2$). The last case is slightly more elaborate. Using that $T_1$ contains no long cycle, it follows that each $C\in \mathcal{C}_1$ must have a subpath that traverses $T_1$ from $t_1$ to $v^*$. Since there are at most $p_1$ edge-disjoint such paths, it follows that $|\mathcal{C}_1| \leq p_1 \leq p_2$. In turn, this implies that we can associate each $C\in \mathcal{C}_1$ with a unique subpath $P_C$ of $T_2$ that joins $v$ and $t_2$, such that these paths are edge-disjoint. Now fix $C\in \mathcal{C}_1$. The decontraction of $v^*$ naturally maps $C$ to a path $P$ (on the same edges) which joins $v$ and $t_2$ in $G-E(T_2)$. Since $P$ and $P_C$ have the same endpoints and are internally vertex-disjoint, we can combine them into a cycle; we define $\sigma(C)$ to be that cycle. This concludes the definition of the mapping $\sigma$. Note that for all $C$, the length of $\sigma(C)$ is at least the length of $C$, so we indeed obtain a collection of long cycles. Moreover, edge-disjointness is preserved because we did not modify the edge-sets outside $T_2$, while inside $T_2$ we added edge-disjoint paths.\
Second, suppose that $G^*-F$ has no long cycles, for some $F\subseteq E(G^*)$ 