and only if the point $({\mathtt{w}}(u),{\mathtt{w}}(v))$ lies in the slab.

![(Left) A double-threshold graph. The weight of each vertex is given as ${\mathtt{w}}(a)=1$, ${\mathtt{w}}(b)=3$, ${\mathtt{w}}(c)=5$, and ${\mathtt{w}}(d)=7$; the lower bound is ${\mathtt{lb}}=4$ and the upper bound is ${\mathtt{ub}}=8$. (Right) Understanding the definition in the $xy$-plane. A white dot represents the point $({\mathtt{w}}(u),{\mathtt{w}}(v))$ for distinct vertices $u,v$, and a cross represents a point $({\mathtt{w}}(v),{\mathtt{w}}(v))$ for a vertex $v$. Two distinct vertices $u$ and $v$ are joined by an edge if and only if the corresponding white dot lies in the gray slab. []{data-label="fig:dtg_example1"}](\figdir/dtg_example1)

Every threshold graph is a double-threshold graph as one can set a dummy upper bound ${\mathtt{ub}}> \max \{{\mathtt{w}}(u) + {\mathtt{w}}(v) \mid u,v \in V\}$. From the definition of double-threshold graphs, we can easily see that they coincide with the star pairwise compatibility graphs.

A graph is a double-threshold graph if and only if it is a star pairwise compatibility graph.

Let $G = (V,E)$ be a double-threshold graph defined by ${\mathtt{w}}\colon V \to {\mathbb{R}}$ and ${\mathtt{lb}}, {\mathtt{ub}}\in {\mathbb{R}}$. We construct an edge-weighted star $S$ with the center $c$ and the leaf set $V$ such that the weight ${\mathtt{w}}'(vc)$ of each edge $vc \in E(S)$ is ${\mathtt{w}}(v)$. Then, $G$ 