Bounding the Number of Minimal Dominating Sets: A Measure and Conquer Approach

We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697n, thus improving on the trivial $\mathcal{O}(2^{n}/\sqrt{n})$ bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an $\mathcal{O}(1.7697^{n})$ listing algorithm. Based on this result, we derive an $\mathcal{O}(2.8805^{n})$ algorithm for the domatic number problem, and an $\mathcal{O}(1.5780^{n})$ algorithm for the minimum-weight dominating set problem. Both algorithms improve over the previous algorithms.

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