Is there a math term for a modificated minimum spanning tree?

Question

I have found a minimum spanning tree (left figure). Then I have applied some modification:

1. The edge A-B (red) was added,
2. The edge C-D (green) was rewritten to C'-D.

Edit. I don't delete the vertices.

3. Steps 1-2 can be repeated more that one time for different edges.

Question. Is there a math term for an obtained graph (right figure)?

Answer 1

Well...it's no longer a tree, so I guess the best I can say is that it's a spanning subgraph. I'm not sure whether your "modifications" always preserve the property that the new graph contains all the vertices. If not, then you can just say "it's a subgraph."

Answer 2

There is a term for this subgraph with a unique cycle: it is a $1$-tree.

Some authors also call it a pseudotree.