As stated in the comment, there are a couple of interpretations of “all node cuts of minimal size consisting of only nodes from one set in a bipartite graph”. It either means

- All node cuts of minimum size when restricting cuts to be in one set of the bipartite graph, or
- All node cuts in an unconstrained sense (consisting of nodes from A or B) that happen to completely lie in B.

From your code example you are interested in 2. According to the docs, there is a way to speed up this calculation, and from profile results it helps a bit. There are auxiliary structures built, per graph, to determine the minimum node cuts. Each node is replaced by 2 nodes, additional directed edges are added, etc. according to the Algorithm 9 in http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
We can reuse these structures instead of reconstructing them inside a tight loop:

Improvement for Case 2:

```
from networkx.algorithms.connectivity import (
build_auxiliary_node_connectivity)
from networkx.algorithms.flow import build_residual_network
from networkx.algorithms.flow import edmonds_karp
def getone_sided_cuts_Case2(G, A, B):
# build auxiliary networks
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, 'capacity')
# get all cutes that consist of nodes exclusively from B which disconnet
# nodes from A
one_sided_cuts = []
seen = []
l = list(combinations(A,2))
for x in l:
s = x[0]
t = x[1]
cut = minimum_st_node_cut(G, s, t, auxiliary=H, residual=R)
if set(cut).issubset(B):
if cut not in seen:
one_sided_cuts.append(cut)
seen.append(cut)
# Find minimum cut size
cur_min = float('inf')
for i in one_sided_cuts:
if len(i) < cur_min:
curr_min = len(i)
one_sided_cuts = [x for x in one_sided_cuts if len(x) == cur_min]
return one_sided_cuts
```

For profiling purposes, you might use the following, or one of the built-in bipartite graph generators in Networkx:

```
def create_bipartite_graph(size_m, size_n, num_edges):
G = nx.Graph()
edge_list_0 = list(range(size_m))
edge_list_1 = list(range(size_m,size_m+size_n))
all_edges = []
G.add_nodes_from(edge_list_0, bipartite=0)
G.add_nodes_from(edge_list_1, bipartite=1)
all_edges = list(product(edge_list_0, edge_list_1))
num_all_edges = len(all_edges)
edges = [all_edges[i] for i in random.sample(range(num_all_edges), num_edges)]
G.add_edges_from(edges)
return G, edge_list_0, edge_list_1
```

Using `%timeit`

, the second version runs about 5-10% faster.

For Case 1, the logic is a little more involved. We need to consider minimal cuts from nodes only inside B. This requires a change to `minimum_st_node_cut`

in the following way. Then replace all occurences of `minimum_st_node_cut`

to `rest_minimum_st_node_cut`

in your solution or the Case 2 solution I gave above, noting that the new function also requires specification of the sets `A`

, `B`

, necessarily:

```
def rest_build_auxiliary_node_connectivity(G,A,B):
directed = G.is_directed()
H = nx.DiGraph()
for node in A:
H.add_node('%sA' % node, id=node)
H.add_node('%sB' % node, id=node)
H.add_edge('%sA' % node, '%sB' % node, capacity=1)
for node in B:
H.add_node('%sA' % node, id=node)
H.add_node('%sB' % node, id=node)
H.add_edge('%sA' % node, '%sB' % node, capacity=1)
edges = []
for (source, target) in G.edges():
edges.append(('%sB' % source, '%sA' % target))
if not directed:
edges.append(('%sB' % target, '%sA' % source))
H.add_edges_from(edges, capacity=1)
return H
def rest_minimum_st_node_cut(G, A, B, s, t, auxiliary=None, residual=None, flow_func=edmonds_karp):
if auxiliary is None:
H = rest_build_auxiliary_node_connectivity(G, A, B)
else:
H = auxiliary
if G.has_edge(s,t) or G.has_edge(t,s):
return []
kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H)
for node in [x for x in A if x not in [s,t]]:
edge = ('%sA' % node, '%sB' % node)
num_in_edges = len(H.in_edges(edge[0]))
H[edge[0]][edge[1]]['capacity'] = num_in_edges
edge_cut = minimum_st_edge_cut(H, '%sB' % s, '%sA' % t,**kwargs)
node_cut = set([n for n in [H.nodes[node]['id'] for edge in edge_cut for node in edge] if n not in A])
return node_cut - set([s,t])
```

We then have, for example:

```
In [1]: G = nx.Graph()
# A = [0,1,2,3], B = [4,5,6,7]
In [2]: G.add_edges_from([(0,4),(0,5),(1,6),(1,7),(4,1),(5,1),(6,3),(7,3)])
In [3]: minimum_st_node_cut(G, 0, 3)
{1}
In [4]: rest_minimum_st_node_cut(G,A,B,0,3)
{6, 7}
```

Finally note that the `minimum_st_edge_cut()`

function returns `[]`

if two nodes are adjacent. Sometimes the convention is to return a set of `n-1`

nodes in this case, all nodes except the source or sink. Anyway, with the empty list convention, and since your original solution to Case 2 loops over node pairs in `A`

, you will likely get `[]`

as a return value for most configurations, unless no nodes in `A`

are adjacent, say.

**EDIT**

The OP encountered a problem with bipartite graphs for which the sets A, B contained a mix of integers and str types. It looks to me like the `build_auxiliary_node_connectivity`

converts those str nodes to integers causing collisions. I rewrote things above, I think that takes care of it. I don't see anything in the `networkx`

docs about this, so either use all integer nodes or use the `rest_build_auxiliary_node_connectivity()`

thing above.

`G = nx.Graph()`

`G.add_edges_from([0,4), (0,5), (1,4), (1,5), (1,6), (1,7), (3,6), (3,7)])`

the minimal set of nodes from B gives a size 2 set:`{6, 7}`

or`{4,5}`

while the minimal set is`{1}`

which is in A.