text: # 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
  • We are a question-and-answer site, so we require you to articulate a specific question about your situation. It might also be helpful to share with us the context where you encountered this task. Finally, we discourage questions that simply state an exercise-style task out of context and expect us to solve it. You might find this page helpful in improving your question.
  – 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.
explanation: The provided text is a discussion about reducing the dominating set problem to the satisfiability (SAT) problem. It includes a brief explanation of the problem and a reference to a similar reduction (clique to SAT). The text also contains a comment from a user asking for clarification on the specific question and a response from the original poster clarifying that the question is about how to reduce the dominating set problem to SAT. The text concludes with a detailed explanation of how to perform this reduction.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
  • We are a question-and-answer site, so we require you to articulate a specific question about your situation. It might also be helpful to share with us the context where you encountered this task. Finally, we discourage questions that simply state an exercise-style task out of context and expect us to solve it. You might find this page helpful in improving your question.
  – 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a discussion about reducing the dominating set problem to the satisfiability (SAT) problem. It includes a brief explanation of the problem and a reference to a similar reduction (clique to SAT). The text also contains a comment from a user asking for clarification on the specific question and a response from the original poster clarifying that the question is about how to reduce the dominating set problem to SAT. The text concludes with a detailed explanation of how to perform this reduction.
  
  Label: question-and-answer