def minimum_st_node_cut(G, s, t, aux_digraph=None, mapping=None):
r"""Returns a set of nodes of minimum cardinality that disconnect source
from target in G.
This function returns the set of nodes of minimum cardinality that,
if removed, would destroy all paths among source and target in G.
Parameters
----------
G : NetworkX graph
s : node
Source node.
t : node
Target node.
Returns
-------
cutset : set
Set of nodes that, if removed, would destroy all paths between
source and target in G.
Examples
--------
>>> # Platonic icosahedral graph has node connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_node_cut(G, 0, 6))
5
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of max-flow problems (ie local st-node
connectivity, see local_node_connectivity) to determine the capacity
of the minimum cut on an auxiliary directed network that corresponds
to the minimum node cut of G. It handles both directed and undirected
graphs.
This implementation is based on algorithm 11 in [1]_. We use the Ford
and Fulkerson algorithm to compute max flow (see ford_fulkerson).
See also
--------
node_connectivity
edge_connectivity
minimum_edge_cut
max_flow
ford_fulkerson
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if aux_digraph is None or mapping is None:
H, mapping = _aux_digraph_node_connectivity(G)
else:
H = aux_digraph
edge_cut = minimum_st_edge_cut(H, '%sB' % mapping[s], '%sA' % mapping[t])
# Each node in the original graph maps to two nodes of the auxiliary graph
node_cut = set(H.node[node]['id'] for edge in edge_cut for node in edge)
return node_cut - set([s,t])
def minimum_node_cut(G, s=None, t=None):
r"""Returns a set of nodes of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of nodes of minimum cardinality that, if removed, would destroy
all paths among source and target in G. If not, it returns a set
of nodes of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional (default=None)
t : node
Target node. Optional (default=None)
Returns
-------
cutset : set
Set of nodes that, if removed, would disconnect G. If source
and target nodes are provided, the set contians the nodes that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has node connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_node_cut(G))
5
>>> # this is the minimum over any pair of non adjacent nodes
>>> from itertools import combinations
>>> for u,v in combinations(G, 2):
... if v not in G[u]:
... assert(len(nx.minimum_node_cut(G,u,v)) == 5)
...
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of max-flow problems (ie local st-node
connectivity, see local_node_connectivity) to determine the capacity
of the minimum cut on an auxiliary directed network that corresponds
to the minimum node cut of G. It handles both directed and undirected
graphs.
This implementation is based on algorithm 11 in [1]_. We use the Ford
and Fulkerson algorithm to compute max flow (see ford_fulkerson).
See also
--------
node_connectivity
edge_connectivity
minimum_edge_cut
max_flow
ford_fulkerson
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
# Local minimum node cut
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
return minimum_st_node_cut(G, s, t)
# Global minimum node cut
# Analog to the algoritm 11 for global node connectivity in [1]
if G.is_directed():
if not nx.is_weakly_connected(G):
raise nx.NetworkXError('Input graph is not connected')
iter_func = itertools.permutations
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors_iter(v),
G.successors_iter(v)])
else:
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is not connected')
iter_func = itertools.combinations
neighbors = G.neighbors_iter
# Choose a node with minimum degree
deg = G.degree()
min_deg = min(deg.values())
v = next(n for n,d in deg.items() if d == min_deg)
# Initial node cutset is all neighbors of the node with minimum degree
min_cut = set(G[v])
# Reuse the auxiliary digraph
H, mapping = _aux_digraph_node_connectivity(G)
# compute st node cuts between v and all its non-neighbors nodes in G
# and store the minimum
for w in set(G) - set(neighbors(v)) - set([v]):
this_cut = minimum_st_node_cut(G, v, w, aux_digraph=H, mapping=mapping)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
# Same for non adjacent pairs of neighbors of v
for x,y in iter_func(neighbors(v),2):
if y in G[x]: continue
this_cut = minimum_st_node_cut(G, x, y, aux_digraph=H, mapping=mapping)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
return min_cut
