Characterization and Linear-time Recognition of Paired Threshold Graphs

TL;DR

This paper introduces paired threshold graphs, characterizes them through vertex orderings, and provides a linear-time recognition algorithm, expanding understanding of graph classes that generalize threshold and interval graphs.

Contribution

It offers a new vertex ordering characterization of paired threshold graphs and develops a linear-time recognition algorithm for this class.

Findings

Paired threshold graphs are a subclass of interval graphs.

A vertex ordering characterization of paired threshold graphs is established.

A linear-time recognition algorithm for paired threshold graphs is developed.

Abstract

In a paired threshold graph, each vertex has a weight, and two vertices are adjacent if their weight sum is large enough and their weight difference is small enough. It generalizes threshold graphs and unit interval graphs, both very well studied. We present a vertex ordering characterization of this graph class, which enables us to prove that it is a subclass of interval graphs. Further study of clique paths of paired threshold graphs leads to a simple linear-time recognition algorithm for the class.