Oriented Threshold Graphs

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.

Counting Labeled Threshold Graphs with Eulerian Numbers

“…For example, G is a threshold graph if and only if it contains no induced subgraph isomorphic to 2K 2 , P 4 , or C 4 [6]. Variations such as random threshold graphs [4] and oriented threshold graphs [2] have been studied in recent years. We refer the reader to the book "Threshold Graphs and Related Topics" [6] for more information and characterizations of threshold graphs.…”