Assume I have a directed graph with a start node A and an end node C, and I already know that the shortest path from A to C in this graph is of length `n`

and uses nodes A, B_{0},B_{1}, ..., B_{n-2}, C.

I want to know whether adding a given new edge to this graph will create a new shortest path from A to C of length `< n`

. Of course, I could simply use Dijkstra's algorithm to check the shortest path in this new graph, but my question is whether the information we have about the shortest path in the graph *without* this new edge can somehow be used to solve this problem more efficiently?