Find the minimum number of nodes that need to be removed to make graph disconnected( there exists no path from some node x to all other nodes). Number of nodes can be 105
[Math] Remove minimum number of nodes to make graph disconnected
graph theorynetwork-flow
Best Answer
You are searching for the minimum $k$ such that your graph $G = (V, E)$ is $k$-vertex-connected.
To solve this problem consider Menger's theorem:
(from Wikipedia)
You can find such paths using a reduction of the vertex-version of Menger's theorem to the edge-version of Menger's theorem, which may be solved in a variety of ways, e.g. by solving a max-flow problem.
$k$ is the minimum over the cardinality of a minimum vertex cut for $(u, v)$ for every pair $(u, v)$ of non-adjacent nodes in $G$. $$ k = \min \left\{ \text{cardinality of minimum $u$-$v$-cut} \mid (u, v) \in V \times V, u~\text{and}~v~\text{are non-adjacent} \right\} $$