Finding minimum spanning 1-tree We are considering a connected weighted graph G. 1-tree is a tree with one extra edge added (so it contains exactly one cycle). The task is to find minimum spanning 1-tree of G.
I was thinking of first using any of well-known MST algorithms and then adding an edge of smallest weight to the MST (which is not in the MST). Will something like this work (I am having trouble proving it would)?
I'm not sure this is research level, but here's a sketch. Suppose this algorithm outputs a 1-tree and the actual cheapest 1-tree is some other 1T. Now, since you built your tree with Kruskal's algorithm, you know that the tree directly proceeding adding your last edge(the cycle-inducing edge) was minimal, and as such costs less than any subtree of the "true" cheapest 1-tree.

So, consider the subtree of the "true" 1-tree that is comprised of all edges save one edge in the cycle that your tree doesn't contain(it is easy to show that, unless their cycles are equal, this edge must exist)*. This tree is more expensive than yours. Moreover, since your tree is a tree, adding that edge from the "true" 1T would have made yours a 1-tree. So your final edge must be cheaper than that edge, or you would have chosen that edge. Thus, your 1-tree must be cheapest.

*If the cycles are the same, but the remaining subtrees are different, you are also done since yours was built using Kruskal's algorithm.