How to break up a series of cycles into one connected structure?

I want to convert a graph of loops and line segments into one connected structure without any loops. I can threshold the image and isolate only the loops but then I get stuck.

Here is the image I want to change.

http://imageshack.us/photo/my-images/23/loopsandlines.jpg

Are you looking to create a "spanning tree" ?

Yes that's exactly what we are looking for. I was wondering how to make a spanning tree from the graph given using the maximum number of branchpoints.

I do not know if biograph would be of any use to you: http://www.mathworks.co.uk/matlabcentral/answers/4035-how-to-keep-distance-in-view-of-minspantree-in-biograph