A Note on Non-Dominating Set Partitions in Graphs
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 Ḡ and the minimum number of sets in a non-total dominating set partition of G equals the domination number of Ḡ. 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.
Desormeaux, Wyatt J.; Haynes, Teresa W.; and Henning, Michael A.. 2016. A Note on Non-Dominating Set Partitions in Graphs. Discussiones Mathematicae - Graph Theory. Vol.36(4). 1043-1050. https://doi.org/10.7151/dmgt.1895 ISSN: 1234-3099