A note on the shortest path through a given link in a network

A network, or a graph, can be described by a set of nodes or vertices and a set of links or edges. Let
uv
be two vertices and
xy
be a link in a given undirected network. We show that a shortest path between
u
and
v
through
xy
can be found by transforming the problem into a minimum cost network flow problem. It may be noted that when the network is directed, the problem is
NP
-complete.