Black-and-white threshold graphs

TL;DR

This paper introduces k-threshold graphs, provides recognition algorithms, characterizations by forbidden subgraphs, and explores special subclasses, including their relation to switching classes of threshold graphs.

Contribution

The paper defines k-threshold graphs, develops recognition algorithms, and characterizes these graphs and their subclasses via forbidden induced subgraphs.

Findings

Recognition algorithm for k-threshold graphs runs in O(n^3) time.

k-threshold graphs are characterized by a finite set of forbidden induced subgraphs.

Restricted 2-threshold graphs are equivalent to the switching class of threshold graphs.

Abstract

Let k be a natural number. We introduce k-threshold graphs. We show that there exists an O(n^3) algorithm for the recognition of k-threshold graphs for each natural number k. k-Threshold graphs are characterized by a finite collection of forbidden induced subgraphs. For the case k=2 we characterize the partitioned 2-threshold graphs by forbidden induced subgraphs. We introduce restricted -, and special 2-threshold graphs. We characterize both classes by forbidden induced subgraphs. The restricted 2-threshold graphs coincide with the switching class of threshold graphs. This provides a decomposition theorem for the switching class of threshold graphs.