# How to uniquely determine a min-cut? [duplicate]

There are possibly several min-cuts for the source and target nodes of a graph. I think I can determine the same min-cut for the same graph by putting the following restriction

"if there are several min-cuts, then I select the min-cut consisting of edges closer to the source atom. By saying close to source atom, I mean you start to travel from the source to the target, then an edge, which is in a min-cut you first encounter"

a,b and c are nodes, A, B and C are edges. The most left node is the source and the most right node is the target.

Example (1)

a --A--> b --B--> c

two sets: {A} and {B}. we chose {A}.

Example (2)

a --A--> b --B--> c

a --C--> d --D--> c

Four sets: {A, C}, {A,D},{C,B} and {B,D}. we chose {A, C}.

Example (3)

a --A--> b --B--> c

a --C--> b --D--> c

Two sets: {A,C} and {B,D}, we choose {A,C}.

We can keep trying, but I think all other graphs are combinations of these 3 examples (I may be missing some cases).

(1) So, I want to prove that by doing that way, I alway find a same min-cut for the same graph if I run the algorithm many times.

If I am right, how can I prove (1)?

or I am thinking too much about a very obvious fact? (^&^)

• Try to define "the min-cut consisting of edges closer to the source atom" more formally, as the minimum cut minimizing (or maximizing) some objective function. Then you can try to prove that this minimum (or maximum) is unique. Mar 26, 2017 at 6:18
• @YuvalFilmus Now is it more formal?
– alim
Mar 26, 2017 at 6:25
• No. A more formal statement would be "the minimum cut minimizing the sum of distances of the cut edges from the source", if that's what you mean. Mar 26, 2017 at 7:55
• @YuvalFilmus I see what you mean. thank you.
– alim
Mar 26, 2017 at 17:26