{\mathtt{ub}}' \in {\mathbb{R}}$ with ${\mathtt{lb}}' < {\mathtt{ub}}'$ such that ${\mathtt{w}}$, ${\mathtt{lb}}'$, and ${\mathtt{ub}}'$ define $G$, $\{v \mid {\mathtt{lb}}/2 \le {\mathtt{w}}(v) \le {\mathtt{ub}}/2\} = \{v \mid {\mathtt{lb}}'/2 \le {\mathtt{w}}(v) \le {\mathtt{ub}}'/2\}$, and ${\mathtt{w}}(u) + {\mathtt{w}}(v) \notin \{{\mathtt{lb}}', {\mathtt{ub}}'\}$ for all $(u,v) \in V^{2}$.

Claim \[clm:nicer-rep1\] is illustrated by [ \[fig:claim2-4\]]{}.

![Proof of Claim \[clm:nicer-rep1\]. (Left) There is a pair $(u,v)$ of vertices such that ${\mathtt{w}}(u)+{\mathtt{w}}(v)={\mathtt{ub}}$. The values $\alpha$ and $\beta$ are shown in red and blue, respectively. (Right) The bounds ${\mathtt{lb}}, {\mathtt{ub}}$ are replaced by ${\mathtt{lb}}', {\mathtt{ub}}'$.[]{data-label="fig:claim2-4"}](\figdir/claim2-4)

\[Proof of Claim \[clm:nicer-rep1\]\] If ${\mathtt{w}}(v) \ge {\mathtt{lb}}/2$ for all $v \in V$, then we set ${\mathtt{lb}}'$ to an arbitrary value smaller than ${\mathtt{lb}}$. Similarly, if ${\mathtt{w}}(v) \le {\mathtt{ub}}/2$ for all $v \in V$, then we use any value larger than ${\mathtt{ub}}$ as ${\mathtt{ub}}'$. Since we are done in these cases, we assume that there exists a vertex with weight less than ${\mathtt{lb}}/2$, and there exists a vertex with weight more than ${\mathtt{ub}}/2$.

Let $\alpha, \beta \in {\mathbb{R}}$ be the following values: $$\begin{aligned}
{2}
  \alpha &= \max&\{{\mathtt{w}}(u) + {\mathtt{w}}(v) \mid (u,v) \in V^{2} \text{ such that } {\mathtt{w}}(u) + {\mathtt{w}}(v) < {\mathtt{lb}}\},
  \\
  \beta  &= \min&\{{\mathtt{w}}(u) + {\mathtt{w}}(v) \mid (u,v) \in V^{2} \text{ such 