Linear-Time Recognition of Double-Threshold Graphs

TL;DR

This paper introduces a new characterization of double-threshold graphs, relating them to bipartite permutation graphs, and provides a linear-time recognition algorithm improving upon previous methods.

Contribution

A novel characterization of double-threshold graphs is proposed, enabling a linear-time recognition algorithm that surpasses previous computational efficiency.

Findings

Established a new relation between double-threshold and bipartite permutation graphs.

Developed a linear-time recognition algorithm for double-threshold graphs.

Improved recognition algorithm from cubic to linear time complexity.

Abstract

A graph $G = (V,E)$ is a double-threshold graph if there exist a vertex-weight function $w \colon V \to \mathbb{R}$ and two real numbers $\mathtt{lb}, \mathtt{ub} \in \mathbb{R}$ such that $uv \in E$ if and only if $\mathtt{lb} \le \mathtt{w}(u) + \mathtt{w}(v) \le \mathtt{ub}$. 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 $O(n^{3} m)$ time, where $n$ and $m$ are the numbers of vertices and edges, respectively.