Given graph G(V,E)and two vertices A and B, show how to eliminate the least number of edges so that length of shortest path between A and B become longer.(I need an algorithm)
Run BFS from $A$, and generate a graph $G'=(V,E')$ that consists only of the edges that are on some shortest path from $A$ to $B$ (i.e. $(x,y)\in E'$ iff $d(a,y)=d(a,x)+1$).
And your answer is the minimal $A-B$ cut in $G'$.