Paired Threshold Graphs

TL;DR

This paper introduces Paired Threshold (PT) graphs, a new class of graphs defined by vertex weights and two inequalities, expanding the modeling capabilities beyond traditional threshold graphs.

Contribution

The paper defines PT graphs, provides their characterization, decomposition, forbidden subgraphs, weight assignment method, recognition algorithm, and analyzes their structural properties.

Findings

PT graphs can be recognized in polynomial time.

PT graphs have specific forbidden induced subgraphs.

Structural properties like diameter and clustering coefficient are characterized.

Abstract

Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate this problem, we introduce a new class of graphs termed Paired Threshold (PT) graphs described through vertex weights that govern the existence of edges via two inequalities. One inequality imposes the constraint that the sum of weights of adjacent vertices has to exceed a specified threshold. The second inequality ensures that adjacent vertices have a weight difference upper bounded by another threshold. We provide a conceptually simple characterization and decomposition of PT graphs, analyze their forbidden induced subgraphs and present a method for performing vertex weight assignments on PT graphs that satisfy the defining constraints. Furthermore, we describe a polynomial-time algorithm for recognizing PT graphs. We conclude our exposition with an analysis of the intersection number, diameter and clustering coefficient of PT graphs.