illustrated by [ \[fig:claim2-5\]]{}.

![Proof of Claim \[clm:nicer-rep2\]. (Left) The second smallest weight is shared by two vertices $z$ and $z'$. (Right) The weight of $z$ is changed, but the double-threshold graph does not change.[]{data-label="fig:claim2-5"}](\figdir/claim2-5){width="\textwidth"}

\[Proof of Claim \[clm:nicer-rep2\]\] Let ${\mathtt{w}}'\colon V \to {\mathbb{R}}$ be a vertex-weight function 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, under these conditions, the number $|\{v \mid {\mathtt{w}}'(v) \ne {\mathtt{w}}'(u) \text{ for all } u \in V \setminus \{v\} \}|$ of unique-weight vertices is maximum. Such ${\mathtt{w}}'$ exists since ${\mathtt{w}}$ itself satisfies all the conditions.

Suppose to the contrary that there are distinct vertices $z, z' \in V$ with the same weight ${\mathtt{w}}'(z) = {\mathtt{w}}'(z')$. Let $\alpha, \beta, \gamma$ be defined as follows: $$\begin{aligned}
\alpha &=&
\begin{cases}
\min \{ {\mathtt{w}}'(v) \mid v \in N(z) \} & \text{if } N(z) \ne \emptyset,\\
+\infty & \text{otherwise};
\end{cases}\\
\beta &=&
\begin{cases}
\min \{ {\mathtt{w}}'(v) \mid v \notin N(z), \; {\mathtt{w}}'(z) + {\mathtt{w}}'(v) > {\mathtt{ub}}' \}
 & \text{if } \{v \notin N(z) \mid {\mathtt{w}}'(z) + {\mathtt{w}}'(v) > {\mathtt{ub}}'\} \ne \emptyset,\\
+\infty 