The question is asked first at here. It described what the problem is and a trival greedy algorithm. Also the accepted answer gave a proof of its NP-completeness.
Problem: Given a graph $G(V,E)$, Find a subset of edges $M \subseteq E$. For each edge $(u,v) \in M$ , $\left( \left(d(u) < c\right) \lor \left(d(v) < c\right) \right)$ holds, where $c=3$. $d(u)$ means degree in $M$, i.e., $d(u)=|(u,v) \in M|$. We call this constraint a degree-constraint.
Output: The maximum sized (cardinality of edges) $M$.
The problem on $c=2$ is already $NP-complete$ as shown by Jukka Suomela. There is a $2-approximation$ algorithm for a general graph already. Jukka devised a $O(|V|c)$ DP for this problem when $G$ is restricted on trees.