For a graph and a subset
of the vertex set
, denote by
the set of vertices in
which are adjacent to a vertex in
. If
, 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.