No, the intuition doesn't really scale very well, especially not if you want the optimal solution.

The problem is known as Independent Set and is not only NP-complete, but even W[1]-complete.

There are heuristics you can use, e.g., pick the smallest degree neighbor into your solution, delete the neighborhood, and rinse and repeat. Of course this is not guaranteed to give a good answer.

Another way is to use LP (rounding) for finding the smallest vertex cover. The complement of a vertex cover is an independent set.