Oriented Threshold Graphs

TL;DR

This paper generalizes threshold graphs to oriented graphs, establishing their definitions' equivalence and enumerating them using Fibonacci numbers, thus broadening understanding of these graph classes.

Contribution

It introduces a generalization of threshold graphs to oriented graphs and proves the equivalence of multiple definitions in this new context.

Findings

Definitions of threshold graphs are extended to oriented graphs.

The equivalence of these definitions is established.

The number of oriented threshold graphs relates to Fibonacci numbers.

Abstract

Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has led to many results about threshold graphs and similar structures. We look to generalize this class of graphs to oriented graphs (directed simple graphs.) We give generalizations to four of the most versatile definitions and show their equivalence in the oriented case. We finish with a proof enumerating the number of these oriented threshold graphs which relates to the Fibonacci numbers.