Finding the minimum cost set of nodes so that once removed, the graph is disconnected

Question

It would be great if someone could help me.

I have an undirected graph where every vertex has a weight and where no edges have weights. I want to find a set of nodes with minimum total weight for which their removal makes the graph disconnected. For example, between removing one node with a weight of 10 that make the graph disconnected and removing 2 nodes with a total weight of 6 that make the graph disconnected, the result set should contain those 2 nodes.

Is there any known algorithm for this problem?

Here is what I've done so far. I've made my code using networkx (python). I've already changed my graph to be directed. For instance, for node 1, I consider 1in and 1out node. and I connect 1in to 1out by the weight of node 1. I also add s and t nodes (I'm not sure if it's correct or not). I defined also capacity for each edge in new directed graph.

When run the code, I get this error: NetworkXUnbounded: Infinite capacity path, flow unbounded above.

deg_G = nx.degree(G)
max_weight = max([deg for i,deg in deg_G])+1

st_Weighted_Complement_G = nx.DiGraph()
r = np.arange(len(Complement_G.nodes))

nodes = ['s','t']
edges = []
for i in r:
    nIn = (str(i)+'in')
    nOut = (str(i)+'out')
    nodes.extend([nIn,nOut])
    edges.extend([(nIn,nOut,{'capacity':deg_G[i],'weight':deg_G[i]}),('s',nIn,{'capacity':math.inf,'weight':0}),
                  (nOut,'t',{'capacity':math.inf,'weight':0})])

for edge in Complement_G.edges:
    print(edge[0],edge[1])
    edges.extend([((str(edge[0]))+'out',(str(edge[1]))+'in',{'capacity':max_weight,'weight':0}),
                  ((str(edge[1]))+'out',(str(edge[0]))+'in',{'capacity':max_weight,'weight':0})])

print(edges)
st_Weighted_Complement_G.add_nodes_from(nodes)
st_Weighted_Complement_G.add_weighted_edges_from(edges)

mincostFlow = nx.max_flow_min_cost(st_Weighted_Complement_G, 's', 't',capacity='capacity',weight='weight')
print(mincostFlow)

Thanks