hY(SrSSKrSSKrSSKJrJr \rSSKJ r J r /SQr \R"SSS.S S \ "S 500S 1S 9SS j5r\R"SSS.S SS00S 1S9SSj5r\RSSj5r\RSSj5rg)z Flow based cut algorithms N)build_residual_network edmonds_karp)!build_auxiliary_edge_connectivity!build_auxiliary_node_connectivity)minimum_st_node_cutminimum_node_cutminimum_st_edge_cutminimum_edge_cut)Gz auxiliary? auxiliarycapacityinf)graphspreserve_edge_attrspreserve_graph_attrsc^^ ^Uc[nUc [T5nOUnSX5S.n[R"XaU40UD6upU un m [ 5n U4SjU 5H"umn U R U U4SjU 55 M$ U $)aReturns the edges of the cut-set of a minimum (s, t)-cut. This function returns the set of edges of minimum cardinality that, if removed, would destroy all paths among source and target in G. Edge weights are not considered. See :meth:`minimum_cut` for computing minimum cuts considering edge weights. Parameters ---------- G : NetworkX graph s : node Source node for the flow. t : node Sink node for the flow. auxiliary : NetworkX DiGraph Auxiliary digraph to compute flow based node connectivity. It has to have a graph attribute called mapping with a dictionary mapping node names in G and in the auxiliary digraph. If provided it will be reused instead of recreated. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See :meth:`node_connectivity` for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. residual : NetworkX DiGraph Residual network to compute maximum flow. If provided it will be reused instead of recreated. Default value: None. Returns ------- cutset : set Set of edges that, if removed from the graph, will disconnect it. See also -------- :meth:`minimum_cut` :meth:`minimum_node_cut` :meth:`minimum_edge_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` Examples -------- This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the connectivity package: >>> from networkx.algorithms.connectivity import minimum_st_edge_cut We use in this example the platonic icosahedral graph, which has edge connectivity 5. >>> G = nx.icosahedral_graph() >>> len(minimum_st_edge_cut(G, 0, 6)) 5 If you need to compute local edge cuts on several pairs of nodes in the same graph, it is recommended that you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for edge connectivity, and the residual network for the underlying maximum flow computation. Example of how to compute local edge cuts among all pairs of nodes of the platonic icosahedral graph reusing the data structures. >>> import itertools >>> # You also have to explicitly import the function for >>> # building the auxiliary digraph from the connectivity package >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity >>> H = build_auxiliary_edge_connectivity(G) >>> # And the function for building the residual network from the >>> # flow package >>> from networkx.algorithms.flow import build_residual_network >>> # Note that the auxiliary digraph has an edge attribute named capacity >>> R = build_residual_network(H, "capacity") >>> result = dict.fromkeys(G, dict()) >>> # Reuse the auxiliary digraph and the residual network by passing them >>> # as parameters >>> for u, v in itertools.combinations(G, 2): ... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R)) ... result[u][v] = k >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) True You can also use alternative flow algorithms for computing edge cuts. For instance, in dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp` which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package. >>> from networkx.algorithms.flow import shortest_augmenting_path >>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path)) 5 r)r flow_funcresidualc30># UH oTU4v M g7fN).0nr s W/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/connectivity/cuts.py &minimum_st_edge_cut..s1y!!Iysc3:># UHoT;dM TU4v M g7frr)rv non_reachableus rrrsAd=.@fq!fds  )default_flow_funcrnx minimum_cutsetupdate)r strrrHkwargs cut_value partition reachablecutsetnbrsr!r"s` @@rr r sh%  -a 0 $9 SF>>!>> from networkx.algorithms.connectivity import minimum_st_node_cut We use in this example the platonic icosahedral graph, which has node connectivity 5. >>> G = nx.icosahedral_graph() >>> len(minimum_st_node_cut(G, 0, 6)) 5 If you need to compute local st cuts between several pairs of nodes in the same graph, it is recommended that you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for node connectivity and node cuts, and the residual network for the underlying maximum flow computation. Example of how to compute local st node cuts reusing the data structures: >>> # You also have to explicitly import the function for >>> # building the auxiliary digraph from the connectivity package >>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity >>> H = build_auxiliary_node_connectivity(G) >>> # And the function for building the residual network from the >>> # flow package >>> from networkx.algorithms.flow import build_residual_network >>> # Note that the auxiliary digraph has an edge attribute named capacity >>> R = build_residual_network(H, "capacity") >>> # Reuse the auxiliary digraph and the residual network by passing them >>> # as parameters >>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R)) 5 You can also use alternative flow algorithms for computing minimum st node cuts. For instance, in dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp` which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package. >>> from networkx.algorithms.flow import shortest_augmenting_path >>> len(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path)) 5 Notes ----- This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of maximum flow computations 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]_. See also -------- :meth:`minimum_node_cut` :meth:`minimum_edge_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf NmappingzInvalid auxiliary digraph.rrrBAr2)rgraphgetr$ NetworkXErrorhas_edger nodes) r r(r)rrrr*r5r+edge_cutedgenodenode_cuts rrrsv -a 0 ggkk)T*G;<<zz!1::a++ $ KF#1 A&67:,a8HSFSH08JTT d#T#HJ !f Ks$Cc^UbUbUcUb[R"S5eUbNUbKUT;a[R"SUS35eUT;a[R"SUS35e[TXUS9$TR5(aH[R"T5(d[R"S5e[ R nU4SjnOM[R"T5(d[R"S5e[ RnTRn[T5n[US5nX6US.n[TTRS 9n [TU 5n [T5[U"U 55- U 1- H,n [TX40UD6n [U 5[U 5:dM*U n M. U"U"U 5S 5H9upUTU ;aM[TX40UD6n [U 5[U 5:dM7U n M; U $) a, 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 value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- cutset : set Set of nodes that, if removed, would disconnect G. If source and target nodes are provided, the set contains 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() >>> node_cut = nx.minimum_node_cut(G) >>> len(node_cut) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package. >>> from networkx.algorithms.flow import shortest_augmenting_path >>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path) True If you specify a pair of nodes (source and target) as parameters, this function returns a local st node cut. >>> len(nx.minimum_node_cut(G, 3, 7)) 5 If you need to perform several local st cuts among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`minimum_st_node_cut` for details. Notes ----- This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of maximum flow computations 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]_. See also -------- :meth:`minimum_st_node_cut` :meth:`minimum_cut` :meth:`minimum_edge_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf )Both source and target must be specified.node not in graph)rInput graph is not connectedc>[RRTRU5TR U5/5$r) itertoolschain from_iterable predecessors successors)r r s r neighbors#minimum_node_cut..neighborss/??00!..2CQ\\RS_1UV Vr1r)rrrkey)r$r;r is_directedis_weakly_connectedrH permutations is_connected combinationsrMrrmindegreer&len)r r(r)r iter_funcrMr*Rr+r min_cutwthis_cutxys` rr r 1s| !)q}JKK } A:""U1#]#;< < A:""U1#]#;< <"1ai@@ }}%%a((""#AB B**  Wq!!""#AB B** KK  *!,Aq*-A$! DF A188A!A$iG Vc)A,' '1# -&q!9&9 w<3x= (G. )A,* !9 &q!9&9 w<3x= (G + Nr1cUbUbUcUb[R"S5e[U5n[US5nX5US.nUbNUbKX;a[R"SUS35eX ;a[R"SUS35e[ XAU40UD6$UR 5(a[R "U5(d[R"S5e[XRS9n[URU55n[U5n [U 5n [U 5H5n [ XIU XS-40UD6n [U 5[U5::aU nM5M7 U$[R"U5(d[R"S5e[XRS9n[URU55nUH0n[R "XS 9n U R#5nU (dM0 O U$U H,n[ XNU40UD6n [U 5[U5::dM*U nM. U$![a6 [ XIU U S 40UD6n [U 5[U5::aU nGM@GMDf=f) a Returns a set of edges of minimum cardinality that disconnects G. If source and target nodes are provided, this function returns the set of edges of minimum cardinality that, if removed, would break all paths among source and target in G. If not, it returns a set of edges of minimum cardinality that disconnects G. Parameters ---------- G : NetworkX graph s : node Source node. Optional. Default value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- cutset : set Set of edges that, if removed, would disconnect G. If source and target nodes are provided, the set contains the edges that if removed, would destroy all paths between source and target. Examples -------- >>> # Platonic icosahedral graph has edge connectivity 5 >>> G = nx.icosahedral_graph() >>> len(nx.minimum_edge_cut(G)) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package. >>> from networkx.algorithms.flow import shortest_augmenting_path >>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path)) 5 If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge connectivity. >>> nx.edge_connectivity(G, 3, 7) 5 If you need to perform several local computations among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`local_edge_connectivity` for details. Notes ----- This is a flow based implementation of minimum edge cut. For undirected graphs the algorithm works by finding a 'small' dominating set of nodes of G (see algorithm 7 in [1]_) and computing the maximum flow between an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in [1]_. For directed graphs, the algorithm does n calls to the max flow function. The function raises an error if the directed graph is not weakly connected and returns an empty set if it is weakly connected. It is an implementation of algorithm 8 in [1]_. See also -------- :meth:`minimum_st_edge_cut` :meth:`minimum_node_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf rCrr6rDrErFrOrr) start_with)r$r;rrr rRrSrWrXr&edgeslistrYrange IndexErrorrUdominating_setpop)r r(r)rr*r[r+r@r\r=rir^Dr r]s rr r sO@ !)q}JKK *!,Aq*-A$! DF } :""U1#]#;< < :""U1#]#;< <"15f55 }}%%a((""#AB B1((#aggdm$Q JqA '.q(Ea%LSFSx=CL0&G1q!!""#AB B1((#aggdm$D!!!5AAq NA*1=f=H8}G ," A '.q(E!HOOx=CL0&G1 's(/H))7I)(I))NNN)__doc__rHnetworkxr$networkx.algorithms.flowrrr#utilsrr__all__ _dispatchablefloatr rr r rr1rrrsJ W ! $$z5<&@A% A  AH! $$tTl3% G  GTNNb``r1