For a graph and a subset of the vertex set , denote by the set of vertices in which are in or adjacent to a vertex in . If , then is said to be a dominating set (of vertices in ).
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 is not a proper subset of any other dominating set 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"].