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Dominating Set


For a graph G and a subset S of the vertex set V(G), denote by N_G[S] the set of vertices in G which are in S or adjacent to a vertex in S. If N_G[S]=V(G), then S is said to be a dominating set (of vertices in G).

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.

DominatingSet

For example, in the Petersen graph illustrated above, the set S={1,2,9} 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"].


See also

Connected Domination Number, Domatic Number, Domatic Partition, Dominance, Domination Number, Domination Polynomial, Minimal Dominating Set, Minimum Dominating Set, Total Dominating Set

Portions of this entry contributed by Nicolas Bray

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References

Alikhani, S. and Peng, Y.-H. "Introduction to Domination Polynomial of a Graph." Ars Combin. 114, 257-266, 2014.Hedetniemi, S. T. and Laskar, R. C. "A. Bibliography on Dominating Sets in Graphs and Some Basic Definitions of Domination Parameters." Disc. Math. 86, 257-277, 1990.Mynhardt, C. M. and Roux, A. "Irredundance Graphs." 14 Apr. 2020. https://arxiv.org/abs/1812.03382.

Referenced on Wolfram|Alpha

Dominating Set

Cite this as:

Bray, Nicolas and Weisstein, Eric W. "Dominating Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DominatingSet.html

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