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
For example, in the
Petersen graph illustrated above,
since the set
is a minimum dominating set (left figure),
is a minimum total dominating set (right figure).
For any simple graph
with no isolated points, the total domination number and ordinary domination
(Henning and Yeo 2013, p. 17). In addition, if
is a bipartite graph, then
et al. 2017), where denotes the graph
connected graph with vertex count ,
et al. 1980, Henning and Yeo 2013, p. 11).
See also Dominating Set
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References Azarija, J.; Henning, M. A.; and Klavar, 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.
New York: Springer, 2013. Total
Domination in Graphs. Cite this as:
Weisstein, Eric W. "Total Domination Number."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/TotalDominationNumber.html