I am to prove that monotone boolean formula satisfiability checking when at most k variables are set to 1 is an NP-Complete problem. Proving that it is in NP is easy, but I'm having difficulty showing that it is in NP-Hard. I have seen this, but I would rather not use its solution, as the textbook for the course I am self studying does not present Dominating Sets as an NP-Complete problem. The NP-Complete problems presented in my book(Dasgupta, Papadimitriou, and Vazirani) are the following: 3SAT, Traveling Salesman, Longest Path, 3D-Matching, Knapsack, Independent Set (related to Dominating Set from what I understand), Vertex Cover, Clique, Integer Linear Programming, Rudrata Path, and Balanced Cut.
From my understanding of the problem, if you restrict the clauses to 2 literals each, then this directly parallels the vertex cover problem, and is thus NP-hard. I was wondering if there was any way that I could use this fact to say that the unrestricted problem is also NP-Hard. I intuitively feel that this should work, because the restricted version is a subset of the restricted version, so in at least some cases the problem is NP-hard. Since you always take the worst case scenario, could you extend this to NP-hardness for the unrestricted problem? If so, how would I word that (this is for a self study, not to be turned in), in the proper form? Because my above explanation has a roundabout method of showing the point.
tl;dr Can you prove that something is np-hard by showing that a subproblem is np-hard? I intuitively feel like you should, but want to confirm.