Given $m$ shortest paths between any two vertices of a graph. Determining whether we can pick $k$ shortest paths such that their union covers all edges.I am trying to reduce set cover problem to it. For a set cover problem we can convert its elements into edges and the set represent a path from the first element to the last one. This conversion of set cover into a graph gives a tree kind of structure. Since between any two nodes in a tree the path is unique so, it has to be shortest as well. I formed a counter example of my own reduction which is given here. I believe that I am making some mistake somewhere but unable to recognize it.
Proving NP Completeness
complexity theory;reductions;np hard;decision problem