A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is a non-dominating (non-total dominating) set partition. We show that the minimum number of sets in a non-dominating set partition of a graph G equals the total domination number of its complement G̅ and the minimum number of sets in a non-total dominating set partition of G equals the domination number of G̅ . This perspective yields new upper bounds on the domination and total domination numbers. We motivate the study of these concepts with a social network application.
@article{bwmeta1.element.doi-10_7151_dmgt_1895,
author = {Wyatt J. Desormeaux and Teresa W. Haynes and Michael A. Henning},
title = {A Note on Non-Dominating Set Partitions in Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {1043-1050},
zbl = {1350.05120},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1895}
}
Wyatt J. Desormeaux; Teresa W. Haynes; Michael A. Henning. A Note on Non-Dominating Set Partitions in Graphs. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 1043-1050. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1895/