The last of the three similar graph problems, it’s also probably the easiest reduction. When I teach NP-Completeness, I’ve had a lot of success starting with this reduction (instead of building a stable of NP-Complete problems from zero like the textbooks do). You kind of have to say “Trust me, Vertex Cover is NP-Complete, I’ll show you why later”, but starting with a simple reduction means that they can follow the idea and the process right from the beginning without having to be bogged down in ugly conversions involving SAT.

**The problem:** Independent Set (IS)

**The definition** Given a graph G=(V,E) and an integer J. Does G have a collection of at least J vertices where *no* edges connect any two vertices in J?

**Example: **Using the same graph:

{a,h,k} is an independent set of size 3.

**The reduction:** From VC. Use the exact same graph, and set J=|V|-K. (G has an independent set of J vertices exactly when it has a vertex cover of K vertices. Vertices not in the cover can’t have edges between them).

**Difficulty**: 1. There’s a reason why I do this first (though sometimes I go from Clique instead). In general, though, I try to avoid making them do problems that are *too* easy for homeworks and tests. If the reduction is trivial, it makes the showing the conversion solves the original problem difficult. You get a lot of proofs that basically say “C’mon! They’re the same thing!”. Which is not what I’m looking for 🙂