charge among the vertices according to the following discharging rules, applied to each pair of vertices $u, v \in V(\operatorname{ct}(G))$:

1.  \[2vtx\] If $u$ is a 2-vertex and $v \in N_3(u)$, then $v$ sends charge $\frac{1}{14}$ to $u$.

2.  \[3vtx\] If $v$ is a 3-vertex with $|\operatorname{Resp}(v)|\leq 10$ and $u \in N_3(v)$, then

    (a) \[adj\] if $d(u,v)=1$ and $|\operatorname{Resp}(u)| = 14$, then $v$ sends charge $\frac{1}{7}$ to $u$;

    (b) \[near\] otherwise, if $d(u,v) \leq 4$, then $v$ sends charge $\frac{1}{14}$ to $u$.

We will now verify the assertion that each vertex has final charge at least $2 + \frac{1}{7}$. If $v$ is a 2-vertex, then since $|N_3(v)| = 2$ the final charge on $v$ is $2 + \frac{1}{7}$ after by Rule R\[2vtx\]. Let $v$ be a 3-vertex. If $u \in N_3(v)$, then $d(u,v) \leq 8$ by Lemma \[lma:caterpillar\]. Claim \[clm:Reducible\] implies that $|\operatorname{Resp}(v)|\leq 14$.

*Case : $|\operatorname{Resp}(v)| \in \{11, 12\}$.*

In this case, $v$ only loses charge by Rule R\[2vtx\], so the final charge is at least $3 - \frac{12}{14} = 2 + \frac{1}{7}$.

*Case : $|\operatorname{Resp}(v)| = 14$.*

By Claim \[clm:Reducible\], the $Y$-type configuration about $v$ is $Y(0,7,7)$. Thus, some vertex $u_1 \in N_3(v)$ is at distance one from 