Question about edge disjoint path

Question

I'm studying about edge disjoint path.

If there is $3$ distinct vertices $(u,v,w)$ in given Graph $G = (V,E)$,

Let there is $u \to v$ has $k$ $(k>1)$ edge disjoint paths, and $v \to w$ has $k$ edge disjoint paths, then can we say that $u \to w$ has $k$ edge disjoint paths?

In my opinion, if there is $u \to v$ edge disjoint paths has a one path which is $w$ is internal node of that path, then we cannot sure that argument.

Is there any suggestion or disagreement about my opinion?

Answer 1

Hint:

• If you can use network flows, then set all the edge capacities to $1$ and observe that you have two flows of size at least $k$ from $u$ to $v$ and from $v$ to $w$. Can you say anything about a flow from $u$ to $w$?
• Another approach is via Menger theorem, that is, by removing $(k-1)$ edges from the graph you can destroy at most $(k-1)$ paths from $u$ to $v$ and at most $(k-1)$ paths from $v$ to $w$. Hence, there is still at least one path from $u$ to $w$. In other words, $u$ and $w$ are at least $k$-edge connected.

I hope this helps $\ddot\smile$