### 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.

Original language | English (US) |
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Pages (from-to) | 291-308 |

Number of pages | 18 |

Journal | Discrete Applied Mathematics |

Volume | 250 |

DOIs | |

State | Published - Dec 11 2018 |

### Keywords

- Forbidden induced subgraphs
- Polynomial-time graph recognition algorithms
- Threshold graphs
- Unit interval graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*250*, 291-308. https://doi.org/10.1016/j.dam.2018.05.008