Please consider a tree graph. There is one unique path connecting any two vertices.


However, I wonder how to address the following question:


Starting from a generic tree, is there an algorithmic way to connect any two nodes of the tree with $M$ unique paths such that no two paths contain the same edge? How can the total number of edges in the modified graph be kept a minimum?


I trying to answer this from a network-engineer's perspective. Any help is appreciated.



Modifying tree Graphs
Yes. See the paper [Minimum augmentation of a tree to a K-edge-connected graph](http://onlinelibrary.wiley.com/doi/10.1002/net.3230180104/abstract) by Ueno, Kajitani and Wada.


