A total dominating set of a graph 
is a subset 
of its vertex set 
such that every graph vertex
has a neighbor in 
, including the members of 
themselves. If ![N_G^t[S^t]](/images/equations/TotalDominatingSet/Inline6.svg)
denotes the set of vertices adjacent to a graph
vertex in 
, this condition is ![N_G^t[S^t]=V(G)](/images/equations/TotalDominatingSet/Inline8.svg)
. A graph with
an isolated vertex has no total dominating set.
A total dominating set differs from an ordinary dominating set because membership in 
does not by itself dominate a graph
vertex. Every member must be adjacent to
another member.
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.
An efficient open dominating set, also called a perfect total dominating set, is a total dominating set 
for which every graph vertex
has exactly one neighbor in 
(Kwon and Lee 2014). Equivalently, the open
graph neighborhoods of the members of 
partition the vertex set. The
induced subgraph on 
is a disjoint union of edges.
In a 
-regular graph on 
vertices, counting neighbor
incidences gives 
, so such a set attains the
lower bound 
.
