Let G $=($V$,$ E$)$ be a flow network with integer capacities. Let e be an edge in E$,$ and let G$$ be a flow network that is the same as G except that the capacity of edge e is one lower in G$$ than in G$.($Thus$,$ the capacity of edge e in G is at least $1).$ Design and analyze an O$(|$E$|+|$V$|)-$time algorithm that takes as input an integer maximum flow f in G$,$ and computes an integer maximum flow f$$ in G$.$ Provide proof of correctness and time complexity.

Hints: Don$$t forget to consider the case where edge e is included in a simple directed cycle. Utilizing some form of BFS$/$DFS will be helpful.