# NP Reduction - Dominating set to SAT

Given a graph G and an integer k , recognize whether G contains dominating set X with no more than k vertices. And that is by finding a propositional formula ϕG,k that is only satisfiable if and only if there exists a dominating set with no more than k vertices, so basically reducing it to SAT-Problem.

The solution to this problem is supposed to be similar to reducing clique to SAT. Here is how that looks like: https://blog.computationalcomplexity.org/2006/12/reductions-to-sat.html

• What is your question? Jun 11 at 13:32
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– D.W.
Jun 11 at 20:20
• Right, sorry. The question is how to do it? How to reduce the dominating set problem to SAT Jun 11 at 21:07

Take a look at the Cook Levin theorem, that shows a reduction from any $$NP$$ problem to $$SAT$$. Since $$Dominating-Set\in NP$$, this is a reduction $$Dominating-Set\le_p SAT$$.

In the case I didn't understand your question and you wanted a reduction $$SAT\le_p Dominating-Set$$, consider creating the following graph: add a node for each variable $$x_i$$ and add another node for its negation $$\overline{x_i}$$, and add an edge between the two nodes. For each such variable, add another node $$h_i$$ that has an edge to $$x_i$$ and $$\overline{x_i}$$.

Additionally, for each clause add a node $$c_j$$ and add an edge between it to any literal in it.

Now we want to find a dominating set of size $$k$$, where $$k$$ is the number of variables in the SAT instance.