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# Minimal Dominating Set

A minimal dominating set is a dominating set in a graph that is not a proper subset of any other dominating set.

Every minimum dominating set is a minimal dominating set, but the converse does not necessarily hold.

Minimal dominating sets can be used to compute the domatic number of a graph.

A dominating set is minimal dominating iff it is irredundant (Mynhardt and Roux 2020).

If a set is dominating and irredundant, it is maximal irredundant and minimal dominating (Mynhardt and Roux 2020).

Domatic Number, Domination Number, Dominating Set, Minimum Dominating Set

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## References

Burger, A. P.; Cockayne, E. J.; and Mynhardt, C. M. "Domination and Irredundance in the Queens' Graph." Disc. Math. 163, 47-66, 1997.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.

## Cite this as:

Weisstein, Eric W. "Minimal Dominating Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinimalDominatingSet.html