## Minimal dominating sets in graph classes: combinatorial bounds and enumeration

*Jean-François Couturier, Pinar Heggernes, Pim van 't Hof and Dieter Kratsch*

*Theoretical Computer Science*, vol. 487, pp. 82-94, 2013.

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A preliminary version of this paper appeared in the proceedings of
SOFSEM 2012, the 38th International Conference on Current Trends in Theory and Practice of Computer Science (held on January 21-27, 2012 in Špindlerův Mlýn, Czech Republic), *Lecture Notes in Computer Science*, vol. 7147, pp. 202-213, 2012.

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### Abstract:

The maximum number of minimal dominating sets that a graph on *n* vertices can have is known to be at most 1.7159^{n}. This upper bound might not be tight, since no examples of graphs with 1.5705^{n} or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the maximum number of minimal dominating sets in graphs on *n* vertices. In some cases, we provide examples of graphs whose number of minimal dominating sets exactly matches the proved upper bound for that class, thereby showing that these bounds are tight. For all considered graph classes, the upper bound proofs are constructive and can easily be transformed into algorithms enumerating all minimal dominating sets of the input graph.