Normally, a minimum spanning forest of a graph G is defined as the union of minimum spanning trees of each of its components. This definition is a generalization of the minimum spanning tree of a connected graph to a graph with arbitrary number of components. A spanning tree has to be connected (i.e. not a multi-component forest).
What is however the forest F of minimal weight of a weighted graph G, such that every vertex in G has an incident edge in F?
I found a partial answer, this is actually a variant of the minimum edge cover. So, rephrasing the question, what is known about the minimum weight edge-cover of a general weighted graph G?(weights are positive reals assigned to edges and the graphs weight is their sum).