Decide whether you think the following statement is true or false. If it is true, give a short explanation. If it is false, give a counterexample. Let G be an arbitrary flow network with a source s, a sink t, and a positive integer capacity ce on every edge e. Let (A, B) be a mimimum s-t cut with respect to these capacities {ce : e ? E}. Now, suppose we add 1 to every capacity; then, (A, B) will still be a minimum s-t cut with respect to these new capacities {1+ ce : e ? E}.

The given statement for the minimum cut (A, B) as per the new capacities is false

Explanation:

Assume a graph whose vertices are s, v1, v2, v3, m, t.

The edges for each i would be (s, vi) and (vi, m).

Also, there is as edge (m, t).

The capacity for the all the edges is 1 except the edge (m, t).

The capacity of the edge (m, t) is 4.

Now set A and B as follows:

A = {s}

B = V – A.

This will result the minimum cut of capacity 3.

But if one is added to the capacity of each edge, then the above cut will result with the capacity of 6.

This 6 is larger than the capacity of 4 which is obtained by the following:

B = {t}

A = V – B.

Thus, the given statement for the minimum cut (A, B) as per the new capacities is false

given the lack of choices, i'm gonna wing it and say that pushing your car off a cliff might not be the best idea. your car might sustain damage from the fall and will probably result in the car being too damaged to jump start.

if that's not good enough, avoid filling the car with bees. bees might get agitated when trapped or during the moving process, and might result in death or painful stings that will make jump starting your car very difficult. but in the event that you do choose to fill your car with bees, remember to wear protective gear!