$c$ of $G'$. Extend the coloring $c$ to color $c(vv_1)$ and $c(vv_2)$ from the colors not in $\{ c(uv), c(uu_1), c(uu_2)\}$, a contradiction. We similarly reach a contradiction when $G_2' = G$.

Therefore, $n(G_i') < n(G)$ and $\operatorname{mad}(G_i') \leq \operatorname{mad}(G)$ for each $i \in \{1,2\}$. Thus, there exist strong 5-edge-colorings $c_1$ and $c_2$ of $G_1'$ and $G_2'$, respectively. For each coloring, the colors on the edges $uv, uu_1, uu_2, vv_1, vv_2$ are distinct. Let $\pi$ be a permutation of the five colors satisfying $\pi(c_2(e)) = c_1(e)$ for each edge $e \in \{uv, uu_1, uu_2, vv_1, vv_2\}$. Then, we extend the coloring $c_1$ of $G_1'$ to all of $G$ by assigning $c_1(e) = \pi(c_2(e))$ for all edges $e \in E(G_2')$. The coloring $c_1$ is a strong 5-edge-coloring of $G$, a contradiction.

If $\operatorname{ct}(G)$ does not have any $3$-vertices, then $\operatorname{ct}(G)$ must be isomorphic to cycle $C_n$. If $n \geq 9$, then $\operatorname{ex}_3(G)$ contains an 8-caterpillar. If $n \in \{5,6,8\}$, then $G$ is a subgraph of $S_5$, $S_6$, or $S_8$, which each has a strong edge-coloring using five colors, discovered by computer. When $n \in\{3,4,7\}$, $G$ does not contain $S_3$, $S_4$, or $S_7$, and any proper subgraph of these graphs is $5$ 