LAUSR

A 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.