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).
