I am interested in the following problem:
Input: a connected undirected graph $G=(V,E)$; a positive weight for each vertex.
Output: a minimum weight subset of $V$ whose removal disconnects $G$.
When each vertex has the same weight this is equivalent to determining the vertex-connectivity. Intuitively, it is the vertex version of the global minimum cut problem.
I can think of an $O(mn^3)$ algorithm that solves $O(n^2)$ minimum $s$-$t$ cut problems, but I expect that there should be a faster approach. Does this problem have a name? What is the fastest known algorithm?