# Show that splitting an edge in a flow network yields an equivalent network.

Need help with this question from my Intro to Algorithms book:

Show that splitting an edge in a flow network yields an equivalent network. More formally, suppose that flow network $G$ contains edge $(u,v)$, and we create a new flow network $G'$ by creating a new vertex $x$ and replacing $(u,v)$ by new edges $(u,x)$ and $(x,v)$ with $c(u,x) = c(u,v)$. Show that a maximum flow in $G'$ has the same value as a maximum flow in $G$.

We need to show each flow in $$G$$ corresponds to a flow in $$G'$$ and vice versa.
One direction: Suppose we have some flow in $$G$$. Since we are only changing one of the edges of an anti-parallel edge-pair, we have an identical flow through all other edges(unchanged edges). Let $$f(u,v)$$ is the flow along edge $$(u,v)$$ in $$G$$. Also note that the capacity of both of the two new edges are same, we can set $$f'(u, v) = f '(u, x) = f'(x, v)$$. Now it is easy to see that this flow is identical to the original one and we also have a valid flow.
Other direction: It is very easy. Here $$f_{in}(x) = f_{out}(x)$$ and there is only one incoming and only one outgoing edge. So, we can remove the vertex $$x$$ and replace the edges $$(u,x)$$ and $$(x,v)$$ with edge $$(u,v)$$ with flow $$f'(u,v)=f'(u,x)=f'(x,v)$$.
So, each flow in $$G$$ corresponds to a flow in $$G'$$ and each flow in $$G'$$ corresponds to a flow in $$G$$.