on the interface pairs. Let $S_1$ be the subset of $S$ consisting of vertices not incident with the rings. Let $G_1$ be the graph such that $G_1$ is obtained from $G_0$ by patching on $S_1$ and $G$ is obtained from $G_1$ by patching on $S\setminus S_1$ and framing on the interface pairs. By Corollary \[cor-patch\] and the previous analysis of $T'_n$, the graph $G_1$ satisfies the conclusions of Lemma \[lemma-col-tw\].

Since the vertices of the interface pairs of $G_0$ have degree two, they do not belong to $S$. We define a coloring $\psi'$ of the rings of $G_1$ as follows. Suppose that $C'_1=u_1u'_2u_3u'_4$ is a ring of $G_1$, where $u'_2$ has degree three. Let $\psi'(u_1)=\psi(u_1)$ and $\psi'(u_3)=\psi(u_3)$. If $\psi$ is $u_1$-diagonal on $C_1$, or if $u'_2\not\in S$, $u_2=u'_2$, and $u_4=u'_4$, then let $\psi'(u'_2)=\psi(u_2)$ and $\psi'(u'_4)=\psi(u_4)$. Otherwise, choose $\psi'(u'_2)$ and $\psi'(u'_4)$ so that $\psi'_i$ is $u'_2$-diagonal on $C'_1$ and so that $\psi'(u'_i)=\psi(u_i)$ for all $i\in\{2,4\}$ such that $u_i=u'_i$. Define $\psi'$ on the other ring $C'_2$ of $G_1$ analogously. Note that for $i\in\{1,2\}$, $C'_i$ is strong in $G_1$, and if $C'_i\neq C_i$, then $C_i$ is weak in $G$; hence, $\psi$ is dangerous on $C_i$ if and only if $\psi'$ is dangerous on 