AbstractA graph $$G = (V,E)$$
G
=
(
V
,
E
)
is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$
w
:
V
→
R
and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$
lb
,
ub
∈
R
such that $$uv \in E$$
u
v
∈
E
if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$
lb
≤
w
(
u
)
+
w
(
v
)
≤
ub
. In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$
O
(
n
3
m
)
time, where n and m are the numbers of vertices and edges, respectively.
Threshold Graph Limits and Random Threshold Graphs
