## graphs – Check if a pair of vertices belongs to the min-cut

Given a digraph $$G = (V, A)$$ with a source $$s in V$$ and a sink $$t in V$$, I need to adapt the graph to find out if a pair of vertices $$u in V$$ and $$v in V$$ belongs to the min-cut $$S$$ Between $$s$$ and $$t$$. In other words, a new summit $$a$$ belongs to the min-cut $$S$$ if and only if $$u in S$$ and $$v in S$$. This new vertex must maintain the original max flow of $$s$$ at $$t$$. I tried to create new arcs $$(u, a)$$ and $$(a, v)$$ with endless abilities, but while $$a$$ belongs to $$S$$ when $$u in S$$, it forces immediately $$v$$ To belong to $$S$$ also, which is not necessarily true if the maximum flow was calculated in the original graph $$G$$. So is there a way to force a new node to be in the min-cut set $$S$$ if a pair of vertices belongs to $$S$$?

## ford fulkerson – Min-cut in a zero flow network from source to well

The max-cut min-flow theorem ensures that the minimum cut of a directed network is equal to the maximum throughput. And we can calculate $$S$$ and $$T$$, are disjoint subsets containing the source and receiver nodes of the residual graph, respectively.

What will be $$S$$ and $$T$$ if the maximum flow from the source to the well is equal to 0, there is no path led from $$s$$ at $$t$$? is $$S$$ is going to be the singleton set {s} or its empty set $$phi$$ and $$T$$ to be the set of summits $$V$$?

## In a Min-Cut problem, how often do I have to run Karger's algorithm in a given set of nodes?

Question:

On 22 nodes, how many times do I have to run Karger's algorithm to find a min-cut problem in an error range (for example, <2 or between 0 and 1)?

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• $$(A_ {i}, B_ {i}) = big ((A_ {1}, B_ {1}), …, (A_ {t}, B_ {t}) big) forall i in Bbb {R}$$
• $$P big ((A_ {i}, B_ {i}) big) geq frac {1} { begin {pmatrix} n \ 2 end {pmatrix}} = p$$, which I interpret as the lower limit of the probability of having at least a minimal reduction.
• A: for each graph $$G$$ with $$n$$ knots and every min cut $$(A, B)$$ (I'm assuming the same thing as $$(A_ {i}, B_ {i})$$) $$P big ((A, B) big) geq p$$.
• B There is a graph $$G$$ with $$n$$ knots and a min cut $$(A, B)$$ (still assuming the same thing as $$(A_ {i}, B_ {i})$$) of $$G$$ such as $$P big ((A, B) big) leq p$$.