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A Note on Non-Dominating Set Partitions in Graphs

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Desormeaux W. J., Haynes T. W., Henning M. A.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

1043-1050

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.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Desormeaux W. J.

1 Haynes T. W.

1 Henning M. A.

1 2016

Wyniki wyszukiwania

A Note on Non-Dominating Set Partitions in Graphs