Help understand a pathological example for the Ford Fulkerson Algorithm Consider the graph below. All edges are undirected and have capacity $\frac{1}{1-\sigma}$ where $\sigma = \frac{\sqrt{5}-1} {2}$ except for the ones indicated otherwise (those in red). I further understand that $\sigma^{n+2} = \sigma^{n}-\sigma^{n+1}$.
I was told that the Fulkerson Algorithm does not terminate on it. However, I do not understand why.  Could you explain that to me?
 A: None of the pathological examples for the Ford-Fulkerson algorithm have the property "the algorithm does not terminate for it". All networks have a maximum flow (well, all finite networks with finite capacity on each edge, and a source and sink for the problem to make sense). If we know a maximum flow, it is easy to find a sequence of augmenting paths that will let the Ford-Fulkerson algorithm find it in finitely many steps.
(In this example, the maximum flow is $\frac{4}{1-\sigma}$, and is easy to find in four augmenting steps: just take four edge-disjoint paths of length $3$ from $s$ to $t$ that don't use any of the red edges.)
So pathological examples for Ford-Fulkerson must come with a pathological choice of augmenting paths which is valid at every step, but does not terminate. Essentially, what we're saying is this: "The Ford-Fulkerson algorithm does not fully specify which augmenting path to choose at each step. So it's possible that it will choose these augmenting paths. Then it will never terminate. As a result, we conclude that it's possible for Ford-Fulkerson to never terminate, unless we come up with a rule to pick augmenting paths that avoids outcomes like this one".
Without knowing the pathological choice of augmenting paths, it's hard to reconstruct it. Presumably, however, the bad augmenting paths were specified wherever you found the example.
If not, here is a different example for you to try to understand. It is a $6$-vertex, $8$-edge example that is:

*

*Originally described in "The smallest networks on which the Ford-Fulkerson maximum flow procedure may fail to terminate" by Uri Zwick.

*Elaborated on in these lecture notes on pages 335-336.

*Worked out in excruciating detail in the last section of my lecture notes.

