## Degree constrained edge partitioning (version 2)

Given a graph $G=(V,E)$ with real-valued (positive or negative) weights assigned to its edges, we want to remove a set of edges so that the sum of the remaining edges is minimized and the degree of any vertex should be different than 1 (i.e. 0 or more than 1) in the final graph.

I'm interested in the complexity of this problem.

Note that this problem is a slight variation of this.

-
 So there should be no leaf vertices in the final graph. Isolated vertices are OK. – eakbas Apr 9 2010 at 1:10

 Didn't you show in you answer to the linked question that "minimum subgraph of degree $\ge 2$" is polynomial? – Sergei Ivanov Apr 9 2010 at 8:41 The difference is that in this case one has to decide which vertices to include, and which to give degree zero. That makes it harder. – David Eppstein Apr 9 2010 at 17:34