First off, I would like to clarify that I've seen this: Finding Bottleneck Edges in a Graph in O(V+E)
But I am confused by the answer, as a bottleneck edge is defined as an edge such that increasing its capacity would increase the overall flow.
But the answer seems to say that if multiple edges are needed simultaneously to increase the flow, then they are all bottleneck edges, even if increasing the capacity of any one of them does not result in increased flow?
Consider the example:
S -> a -> b -> T, where S and T are sink and source respectively.
Let k(u, v) = (f, c) be a function denoting the flow and capacity respectively of an edge going from u to v.
Then let's define k(S, a) = k(a, b) = (6, 6), and k(b, T) = (6, 10).
The residual graph would look have flow b -> T and T -> b of 4 and 6 respectively. The residual graph would have no other edges.
In the residual graph, the set of vertices reachable from source S consists of only S itself, and the set of vertices reachable from sink T is {b, T}.
Based on the answer in the question above, both S -> a and a -> b would be bottlenecks, as they (taken in conjunction) connect the first set of vertices with the second set of vertices.
Does this run contrary with the definition that a bottleneck edge is an edge such that increasing its capacity increases the flow, since increasing either S -> a or a -> b alone would not increase the overall flow of the network, or bottleneck edges can form a set collectively and increasing their capacity collectively increases the network flow?