Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
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.
Cite
@article{arxiv.1303.0944,
title = {Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs},
author = {Nina Chiarelli and Martin Milanic},
journal= {arXiv preprint arXiv:1303.0944},
year = {2015}
}
Comments
19 pages, 1 figure