identifying a problem in graph theory

Recently, a friend of mine asked me to help with the following optimization problem in graph theory:

Given a connected graph, decompose it into 3-vertex groups such that the number of resulting groups is minimal. 3-vertex group must be connected in the original graph: it will have 2 or 3 edges. Obviously, there might be some remainder groups with less than 3 vertices. The goal is to minimize the number of such remainder groups as well.

Example: take a map of us. Make states nodes and the borders edges. Take Illinois, Wisconsin, Minnesota, Kentucky, Indiana, Missouri. The solution would produce 2 groups.

We have an algorithm that seems to work. However, I'd like to proove that it works for n nodes. Hence, the question to identify the problem in computer science.

Thank you!

Dmitriy