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: