I am trying to solve a problem and I am really struggling, I would appreciate any help.
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 $\phi_{G,k}$ 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.
What I have so far is this boolean formula:
$\phi=\bigwedge\limits_{i\in V} \bigvee\limits_{j\in V:(i,j)\in E} x_i \vee x_j$
So basically defining a variable $x_i$ that is set to true when when vertix $v_i$ is in the dominating Set $X$, so all the formula says that it is satisfiable when for each node in $G$ it is true that either the vertex itself is in $X$ or one of the adjacent vertices is. Which is basically a Weighted Satisfiability Problem so its satisfiable with at most $k$ variables set to true.
My issue is now that I couldn't come up with a boolean formula $\phi_{G,k}$ that not only uses Graph $G$ as input but also integer $k$. So my question is now how can I modify this formula so it features $k$ , or possibly come up with a new one if it cannot be modified?
