that } {\mathtt{w}}(u) + {\mathtt{w}}(v) > {\mathtt{ub}}\}.\end{aligned}$$ We set ${\mathtt{lb}}' = ({\mathtt{lb}}+ \alpha)/2$ and ${\mathtt{ub}}' = ({\mathtt{ub}}+ \beta)/2$. Then, $\alpha < {\mathtt{lb}}' < {\mathtt{lb}}\le {\mathtt{ub}}< {\mathtt{ub}}' < \beta$, and thus ${\mathtt{lb}}' < {\mathtt{ub}}'$. We can see that ${\mathtt{lb}}'$ and ${\mathtt{ub}}'$ satisfy the other conditions in the claim as follows. Let $(u,v) \in V^{2}$. If ${\mathtt{lb}}\le {\mathtt{w}}(u) + {\mathtt{w}}(v) \le {\mathtt{ub}}$, then ${\mathtt{lb}}' < {\mathtt{w}}(u) + {\mathtt{w}}(v) < {\mathtt{ub}}'$. If ${\mathtt{w}}(u) + {\mathtt{w}}(v) < {\mathtt{lb}}$, then ${\mathtt{w}}(u) + {\mathtt{w}}(v) \le \alpha < {\mathtt{lb}}'$. If ${\mathtt{w}}(u) + {\mathtt{w}}(v) > {\mathtt{ub}}$, then ${\mathtt{w}}(u) + {\mathtt{w}}(v) \ge \beta > {\mathtt{ub}}'$. This implies that ${\mathtt{w}}$, ${\mathtt{lb}}'$, and ${\mathtt{ub}}'$ define $G$, that $\{v \mid {\mathtt{lb}}/2 \le {\mathtt{w}}(v) \le {\mathtt{ub}}/2\} = \{v \mid {\mathtt{lb}}'/2 \le {\mathtt{w}}(v) \le {\mathtt{ub}}'/2\}$, and that ${\mathtt{w}}(u) + {\mathtt{w}}(v) \not\in \{{\mathtt{lb}}',{\mathtt{ub}}'\}$ for all $(u, v) \in V^{2}$.

In the following, let ${\mathtt{lb}}'$ and ${\mathtt{ub}}'$ be the bounds in Claim \[clm:nicer-rep1\].

\[clm:nicer-rep2\] There exists ${\mathtt{w}}' \colon V \to {\mathbb{R}}$ such that ${\mathtt{w}}'$, ${\mathtt{lb}}'$, and ${\mathtt{ub}}'$ define $G$, $\{v \mid {\mathtt{lb}}/2 \le {\mathtt{w}}(v) \le {\mathtt{ub}}/2\} = \{v \mid {\mathtt{lb}}'/2 \le {\mathtt{w}}'(v) \le {\mathtt{ub}}'/2\}$, ${\mathtt{w}}'(u) + {\mathtt{w}}'(v) \notin \{{\mathtt{lb}}', {\mathtt{ub}}'\}$ for all $(u,v) \in V^{2}$, and ${\mathtt{w}}'(u) \ne {\mathtt{w}}'(v)$ if $u \ne v$.

Claim \[clm:nicer-rep2\] is 