So, here is the following question I am trying to answer.
The global edge connectivity of an undirected graph is defined as the minimum number of edges that must be removed to disconnect the graph. Show how to compute the edge connectivity of an undirected graph, $G = (V,E)$ with $|V| = n$, and $|E| = m$ by running the maximum flow algorithm $n-1$ times, each on a flow network with $O(n)$ vertices and $O(m)$ edges.
Here is how I plan to solve this problem.
Since the maximal flow algorithm runs on a network, I assume we already derived a flow network from our graph $G$ with each edge having an equivalent integer capacity (I'm unsure if this assumption would lead to the correct answer). Then, we fix some arbitrary $u \in V$ as the source and iterate over every other vertex in $V$ as a sink. For each vertex, we run the maximum flow algorithm and record the number of augmentations we need for the source and sink to become disconnected. Then, we return the minimum augmentations required to disconnect $u$ from an arbitrary $v \in V\setminus\{u\}$ and that will be our value for the global edge connectivity.
Now, to explain the procedure of making the directed flow network, we take our graph $G$ and define a new graph $G' = (V, E')$ where $E'$ is the "opposite symmetric closure" of the $E$. To explain further, I mean that since our graph was undirected, we define a new graph where we break that symmetry in $E$ and make it into an undirected graph, and we do it the following way. We make sure that each direction of the edge is pointing towards the sink. As in, we greedily direct our edge from our vertex $a$ to a vertex $b$ where the path distance from the sink to vertex $b$ is smaller than vertex $a$.
What I'm unsure about is the correctness of my approach. I doubt that this set of processes will yield the answer I'm looking for. I'm also confused how a data structure that has directed edges will be of any help when reasoning about a data structure with no such direction.
