Total Domination Number
المؤلف:
Azarija, J.; Henning, M. A.; and Klavžar, S.
المصدر:
(Total) Domination in Prisms." Electron. J. Combin. 24, No. 1, paper 1.19, 2017.
الجزء والصفحة:
...
15-3-2022
2207
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 ![S=<span style=]()
{1,2,9}" src="https://mathworld.wolfram.com/images/equations/TotalDominationNumber/Inline4.svg" style="height:22px; width:98px" /> is a minimum dominating set (left figure), while 
since ![S^t=<span style=]()
{4,8,9,10}" src="https://mathworld.wolfram.com/images/equations/TotalDominationNumber/Inline6.svg" style="height:22px; width:133px" /> 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.
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