Weakly threshold graphs

TL;DR

This paper introduces weakly threshold graphs, a new class extending threshold graphs, with properties like majorization, an iterative construction, forbidden subgraph characterization, and exact enumeration.

Contribution

It defines weakly threshold sequences and graphs, extending threshold graph properties, and provides algorithms, characterizations, and enumeration methods for this new class.

Findings

Weakly threshold sequences satisfy a specific inequality involving degree sums.

Weakly threshold graphs can be constructed iteratively.

The paper provides a forbidden subgraph characterization and exact enumeration.

Abstract

We define a weakly threshold sequence to be a degree sequence $d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i \geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq \max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs.