Bounded occurrence 3D matching problem
It is solvable in polynomial time.

Convert an instance of Bounded Occurrence 3DM to a cubic graph.

For each triple, create a vertex

For each element, there are exactly two triple containing it. So, there are exactly two vertices corresponding to these two triples. Connect these two vertices by an edge.

Now, a solution to your problem is an independent vertex cover.

A graph has an independent vertex cover iff. it is bipartite. Checking for bipartiteness can be done in polynomial time.