Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

TL;DR

This paper introduces total domishold graphs, a new class generalizing threshold graphs, with connections to threshold hypergraphs, and provides polynomial recognition algorithms and characterizations.

Contribution

It defines total domishold graphs, explores their properties, and develops recognition algorithms and hereditary characterizations.

Findings

Total domishold graphs properly contain threshold graphs.

Polynomial time recognition algorithm for total domishold graphs.

Characterization of hereditary cases via new hypergraph families.

Abstract

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.