Identifying useless edges for shortest path
If you are looking for a way to name (or alternately characterize) these edges you call "useless" and "necessary," you could refer to them as the edges with betweenness centrality =0 and =1, respectively. Every edge can be classified as having =0, =1 or in(0,1) betweenness measure in time of all-pairs-shortest-paths.

This is a well-studied measure of network edges, and there are fast algorithms for updating all the edges' centrality scores upon edge deletions (but I'm not sure about other perturbations).

A centrality function is built-in to mostly every network analysis I've seen, and there is a definition that applies to directed graphs as well:

(edit: the link I gave initially only discussed node betweenness centrality, but here is the only wikipedia article I can find that discusses edge-betweenness centrality:
http://en.wikipedia.org/wiki/Girvan%E2%80%93Newman_algorithm
Still, edge-betweenness is a standard measure that can usually be found in network analysis packages.)