Total Domination Number

The total domination number  of a graph is the size of a smallest total dominating set, where a total dominating set is a set of vertices of the graph such that all vertices (including those in the set itself) have a neighbor in the set. Total dominating numbers are defined only for graphs having no isolated vertex (plus the trivial case of the singleton graph ).

For example, in the Petersen graph illustrated above,  since the set  is a minimum dominating set (left figure), while  since  is a minimum total dominating set (right figure).

For any simple graph  with no isolated points, the total domination number  and ordinary domination number  satisfy

(1)

(Henning and Yeo 2013, p. 17). In addition, if  is a bipartite graph, then

(2)

(Azarija et al. 2017), where  denotes the graph Cartesian product.

For a connected graph  with vertex count ,

(3)

(Cockayne et al. 1980, Henning and Yeo 2013, p. 11).

REFERENCES

Azarija, J.; Henning, M. A.; and Klavžar, S. "(Total) Domination in Prisms." Electron. J. Combin. 24, No. 1, paper 1.19, 2017.

 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p19.Cockayne, E. J., Dawes, R. M., and Hedetniemi, S. T. "Total Domination in Graphs." Networks 10, 211-219, 1980.

Henning, M. A. and Yeo, A. Total Domination in Graphs. New York: Springer, 2013.