Given a graph $G=(V,E)$ and an integer $k$, is there a subset of vertices with size at most $k$ such that every triangle* in $G$ has at least one vertex contained in this set?
*triangle: a set of three vertexes in which every vertex is connected by an edge to the other two vertexes
I need to prove that this problem is $\mathsf{NP}$-hard using known $\mathsf{NP}$-hard problem. I thought of reduction from $\mathsf{3SAT}$ by converting the $\mathsf{3SAT}$ expression to a graph in which every clause is a triangle and every literal in the clause will be converted to a vertex. I had a problem converting the other direction : given a graph $G = (V,E)$, I have to convert every triangle to a clause, but what about the other vertices not in a triangle? Is the reduction from $\mathsf{3SAT}$ possible?