Question

I am reading about network flow algorithms in Robert Sedwicks book on Graph algorithms. Following is text snippet from the book.

Property: Any st-flow has the property that outflow from s is equal to
the inflow to t.

Proof: Augument the network with an edge from a dummy vertex into s,
with flow and capacity equal to the outflow from “s”, and with an edge
from “t” to another dummy vertex, with flow and capactiy equal to the
inflow to “t”. Then, we can prove a more general property by
induction: Inflow is equal to outflow for any set of vertices (not
including the dummy vertices).

This property is true for any single vertex, by local equilibrium.
Now, assume that it is true for a given set of vertices “S” and that
we add a single vertex “v” to make the set S1 = S U {v}. To compute
inflow and outflow for S1, note that each edge from “v” to some vertex
in S reduces outflow (from V) by the same amount as it reduces inflow
(to S); each edge to v from some vertex in S reduces inflow (to v) by
the same amount as it reduces outflow (from S); and all other edges
provide inflow or outflow for S1 if and only if they do so for S or v.
Thus, inflow and outflow are equal for S1, and the value of the flow
is equal to the sum of the values of the flows of v and S minus sum of
the flows on the edges connectin v to a vertex in S (either
direction).

Applying this property to the set of all the networks vertices, we
find that the source’s inflow from its associated dummy vertex is
equal to the sink’s outflow to tits assicated dummy vertex.

My question on above proof:

1. What does author mean by “that each edge from “v” to some vertex in S reduces outflow (from V) by the same amount as it reduces inflow (to S)” ?
   can any one explain with simple example.

2. What does author mean by “each edge to v from some vertex in S reduces inflow (to v) by the same amount as it reduces outflow (from S);
   and all other edges provide inflow or outflow for S1 if and only if they do so for S or v” ? please explain with simple example.

3. What does author mean by ” inflow and outflow are equal for S1, and the value of the flow
   is equal to the sum of the values of the flows of v and S minus sum of the flows on the edges connectin v to a vertex in S (either direction).” ? pls explain with example .

Thanks!

Answer 1

The paragraph is not very well written but if I get it right in short the author means that flow from v to a vertex from S(name this vertex t) increases the outflow from v with a given value, but it also increases the inflow of t with the same value. So adding such an edge we still have that the summar inflow equals the summar outflow. Similarly for an edge from a vertex from S to v we increse the inflow of v and the outflow of the vertex in S with the same value. All othere edges connect two vertices from S and by the inductive assumption their summar inflow equals their summar outflow. Please note that instead of the term reduces the outflow for vertex v I used “increases the outflow” as it seems to be more natural for me.
I would not dare to guess what does the author mean by the last sentence of the second to last paragraph.

Hope that helps at least a little bit.

I believe there are books where this particular theorem seems to be better explained for instance “Introduction to algorithms” written by Thomas H. Cormen,Charles E. Leiserson,Ronald L. Rivest,Clifford Stein.