Is my idea of incoming/outgoing arcs correct?

Question

I'm reading Jungnickel's Graphs, Networks and Algorithms. I've met the following lemma:

I know that $e^{-}$ are the incoming vertices and $e^{+}$ are the outgoing vertices. Then I've tried to draw a digraph to see the result of the lemma, I've made the following digraph:

$$\begin{matrix} {sa(10)}&{ba(10)}&{db(5)}&{ac(9)}&{ca(6)}\\ {sb(7)}&{bd(7)}&{da(8)}&{}&{cd(4)}\\ {}&{}&{dt(10)}&{}&{ct(5)} \end{matrix}$$

So, what is the sum of the value of the flow of the outgoing arc? How to know what is an outgoing arc? My guess is that if you take $bd$ to be and outgoing arc, then $db$ is the incoming arc.

Also, when doing the sum in the lemma, when I have an arc $da$ and don't have $ad$, then it would be the sum of the value of the flow of $da$ minus the sum of the value of the flow of the arc $ad$ and if it does not exist, then $f(ad)=0.$ Is that correct?

Answer 1

Judging from your question, I think you may be misinterpreting the sums. From left to right, you're summing over:

• The arcs that have "s" as their tail
• The arcs that have "s" as their head
• The arcs that have "t" as their head
• The arcs that have "t" as their tail.

In particular, the sums in the lemma do not involve any arcs that are not incident to s or t. Think of the left side of the equation as "net flow leaving $s$" and the right side as "net flow entering $t$".