A dominating set of a graph 
is a subset 
of its vertex set 
such that every graph vertex
is either in 
or adjacent to a graph
vertex in 
.
Writing ![N_G[S]](/images/equations/DominatingSet/Inline6.svg)
for the set of these vertices,
this condition is ![N_G[S]=V(G)](/images/equations/DominatingSet/Inline7.svg)
.
A dominating set of smallest size is called a minimum dominating set and its size is known as the domination number. A dominating set that does not contain any other dominating set as a proper subset is called a minimal dominating set.
For example, in the Petersen graph illustrated above, the set 
is a dominating set (and, in fact, a minimum
dominating set).
The domination polynomial encodes the numbers of dominating sets of various sizes.
Other variants of the usual dominating set can be defined, including the so-called total dominating set.
If a set is dominating and irredundant, it is maximal irredundant and minimal dominating (Mynhardt and Roux 2020).
A dominating set is minimal dominating iff it is irredundant (Mynhardt and Roux 2020).
Precomputed dominating sets for many named graphs can be obtained in the Wolfram Language using GraphData[graph, "DominatingSets"].
