For a graph 
and a subset 
of the vertex set 
, denote by ![N_G^t[S^t]](/images/equations/TotalDominatingSet/Inline4.svg)
the set of vertices in 
which are adjacent to a vertex in 
. If ![N_G^t[S^t]=V(G)](/images/equations/TotalDominatingSet/Inline7.svg)
, then 
is said to be a total dominating set (of vertices in 
). Because members of a total dominating
set must be adjacent to another vertex, total dominating sets are not defined for
graphs having an isolated vertex.
The total dominating set differs from the ordinary dominating set in that in a total dominating set 
, the members of 
are required to themselves be adjacent to a vertex in 
, whereas is an ordinary dominating
set 
,
the members of 
may be either in 
itself or adjacent to vertices in 
.
For example, in the Petersen graph illustrated above, the set 
is a (minimum) dominating set (left figure), while 
is a (minimum) total dominating set (right figure).
The size of a minimum total dominating set 
is called the total
domination number.
