What is complexity of this max-edge subgraph problem? While discussing the question I had asked here, @NealYoung and I encounter another problem, which is to judge complexity of the problem below:


  Given a connected undirected graph, finding a maximum-size subset of
  the edges such that every vertex has degree at most two.


I have found some paper on complexity of relative problems. Most of them added more constraint on the original one. FOS03 added "with no odd cycles" and proved that it is NP-hard. CTW07 implied that the variant added "with no 3-cycle" is P referring to another paper (which I failed to find). But I have been unable to judge complexity of the original problem. So how to judge it? Thanks.
The generalized problem (such that each vertex $v$ has degree at most $b_v$) is called the $b$-matching problem. Chapter 31 of Schrijver's book is devoted to the subject. It can be solved in strongly polynomial time.

The case that $b_v=2$ for every vertex $v$ has been studied, particularly for providing lower bounds for the traveling salesman problem.