minimum vertex set removal for edge-free graph
The problem you describe is called minimum vertex cover. It is a well-known NP-complete problem, so there is no known algorithm that will always be fast (as in polynomial runtime) and give the optimal solution. However, fast approximation solutions are known, as well as algorithms that, while slow in the worst case on general graphs, are fast on significant classes of graphs. See the linked Wikipedia article for details and links to even more details.