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<h1>Source code for networkx.algorithms.connectivity.connectivity</h1><div class="highlight"><pre>
<span class="c"># -*- coding: utf-8 -*-</span>
<span class="sd">"""</span>
<span class="sd">Flow based connectivity algorithms</span>
<span class="sd">"""</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="n">__author__</span> <span class="o">=</span> <span class="s">'</span><span class="se">\n</span><span class="s">'</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">'Jordi Torrents <jtorrents@milnou.net>'</span><span class="p">])</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span> <span class="s">'average_node_connectivity'</span><span class="p">,</span>
<span class="s">'local_node_connectivity'</span><span class="p">,</span>
<span class="s">'node_connectivity'</span><span class="p">,</span>
<span class="s">'local_edge_connectivity'</span><span class="p">,</span>
<span class="s">'edge_connectivity'</span><span class="p">,</span>
<span class="s">'all_pairs_node_connectivity_matrix'</span><span class="p">,</span>
<span class="s">'dominating_set'</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="average_node_connectivity"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.average_node_connectivity.html#networkx.algorithms.connectivity.connectivity.average_node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">average_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">r"""Returns the average connectivity of a graph G.</span>
<span class="sd"> The average connectivity `\bar{\kappa}` of a graph G is the average </span>
<span class="sd"> of local node connectivity over all pairs of nodes of G [1]_ .</span>
<span class="sd"> .. math::</span>
<span class="sd"> \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : float</span>
<span class="sd"> Average node connectivity</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> local_node_connectivity</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> local_edge_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average </span>
<span class="sd"> connectivity of a graph. Discrete mathematics 252(1-3), 31-45.</span>
<span class="sd"> http://www.sciencedirect.com/science/article/pii/S0012365X01001807</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">num</span> <span class="o">=</span> <span class="mf">0.</span>
<span class="n">den</span> <span class="o">=</span> <span class="mf">0.</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">den</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">num</span> <span class="o">+=</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">)</span>
<span class="k">if</span> <span class="n">den</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c"># Null Graph</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">num</span><span class="o">/</span><span class="n">den</span>
</div>
<span class="k">def</span> <span class="nf">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">r""" Creates a directed graph D from an undirected graph G to compute flow</span>
<span class="sd"> based node connectivity.</span>
<span class="sd"> For an undirected graph G having `n` nodes and `m` edges we derive a </span>
<span class="sd"> directed graph D with 2n nodes and 2m+n arcs by replacing each </span>
<span class="sd"> original node `v` with two nodes `vA`,`vB` linked by an (internal) </span>
<span class="sd"> arc in D. Then for each edge (u,v) in G we add two arcs (uB,vA) </span>
<span class="sd"> and (vB,uA) in D. Finally we set the attribute capacity = 1 for each </span>
<span class="sd"> arc in D [1].</span>
<span class="sd"> For a directed graph having `n` nodes and `m` arcs we derive a </span>
<span class="sd"> directed graph D with 2n nodes and m+n arcs by replacing each </span>
<span class="sd"> original node `v` with two nodes `vA`,`vB` linked by an (internal) </span>
<span class="sd"> arc `(vA,vB)` in D. Then for each arc (u,v) in G we add one arc (uB,vA) </span>
<span class="sd"> in D. Finally we set the attribute capacity = 1 for each arc in D.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and </span>
<span class="sd"> Erlebach, 'Network Analysis: Methodological Foundations', Lecture </span>
<span class="sd"> Notes in Computer Science, Volume 3418, Springer-Verlag, 2005. </span>
<span class="sd"> http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf</span>
<span class="sd"> </span>
<span class="sd"> """</span>
<span class="n">directed</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">()</span>
<span class="n">mapping</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">node</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">mapping</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
<span class="n">D</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'</span><span class="si">%d</span><span class="s">A'</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span><span class="nb">id</span><span class="o">=</span><span class="n">node</span><span class="p">)</span>
<span class="n">D</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'</span><span class="si">%d</span><span class="s">B'</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span><span class="nb">id</span><span class="o">=</span><span class="n">node</span><span class="p">)</span>
<span class="n">D</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'</span><span class="si">%d</span><span class="s">A'</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="s">'</span><span class="si">%d</span><span class="s">B'</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="n">capacity</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">edges</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">():</span>
<span class="n">edges</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="s">'</span><span class="si">%s</span><span class="s">B'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="s">'</span><span class="si">%s</span><span class="s">A'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">target</span><span class="p">]))</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">directed</span><span class="p">:</span>
<span class="n">edges</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="s">'</span><span class="si">%s</span><span class="s">B'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">target</span><span class="p">],</span> <span class="s">'</span><span class="si">%s</span><span class="s">A'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">source</span><span class="p">]))</span>
<span class="n">D</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span><span class="n">edges</span><span class="p">,</span> <span class="n">capacity</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">D</span><span class="p">,</span> <span class="n">mapping</span>
<div class="viewcode-block" id="local_node_connectivity"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.local_node_connectivity.html#networkx.algorithms.connectivity.connectivity.local_node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Computes local node connectivity for nodes s and t.</span>
<span class="sd"> Local node connectivity for two non adjacent nodes s and t is the</span>
<span class="sd"> minimum number of nodes that must be removed (along with their incident </span>
<span class="sd"> edges) to disconnect them.</span>
<span class="sd"> This is a flow based implementation of node connectivity. We compute the</span>
<span class="sd"> maximum flow on an auxiliary digraph build from the original input</span>
<span class="sd"> graph (see below for details). This is equal to the local node </span>
<span class="sd"> connectivity because the value of a maximum s-t-flow is equal to the </span>
<span class="sd"> capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node</span>
<span class="sd"> aux_digraph : NetworkX DiGraph (default=None)</span>
<span class="sd"> Auxiliary digraph to compute flow based node connectivity. If None</span>
<span class="sd"> the auxiliary digraph is build.</span>
<span class="sd"> mapping : dict (default=None)</span>
<span class="sd"> Dictionary with a mapping of node names in G and in the auxiliary digraph.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> local node connectivity for nodes s and t</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph has node connectivity 5 </span>
<span class="sd"> >>> # for each non adjacent node pair</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.local_node_connectivity(G,0,6)</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of node connectivity. We compute the</span>
<span class="sd"> maximum flow using the Ford and Fulkerson algorithm on an auxiliary digraph </span>
<span class="sd"> build from the original input graph:</span>
<span class="sd"> For an undirected graph G having `n` nodes and `m` edges we derive a </span>
<span class="sd"> directed graph D with 2n nodes and 2m+n arcs by replacing each </span>
<span class="sd"> original node `v` with two nodes `v_A`, `v_B` linked by an (internal) </span>
<span class="sd"> arc in `D`. Then for each edge (`u`, `v`) in G we add two arcs </span>
<span class="sd"> (`u_B`, `v_A`) and (`v_B`, `u_A`) in `D`. Finally we set the attribute </span>
<span class="sd"> capacity = 1 for each arc in `D` [1]_ .</span>
<span class="sd"> For a directed graph G having `n` nodes and `m` arcs we derive a </span>
<span class="sd"> directed graph `D` with `2n` nodes and `m+n` arcs by replacing each </span>
<span class="sd"> original node `v` with two nodes `v_A`, `v_B` linked by an (internal) </span>
<span class="sd"> arc `(v_A, v_B)` in D. Then for each arc `(u,v)` in G we add one arc </span>
<span class="sd"> `(u_B,v_A)` in `D`. Finally we set the attribute capacity = 1 for </span>
<span class="sd"> each arc in `D`.</span>
<span class="sd"> This is equal to the local node connectivity because the value of </span>
<span class="sd"> a maximum s-t-flow is equal to the capacity of a minimum s-t-cut (Ford </span>
<span class="sd"> and Fulkerson theorem).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> all_pairs_node_connectivity_matrix</span>
<span class="sd"> local_edge_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and </span>
<span class="sd"> Erlebach, 'Network Analysis: Methodological Foundations', Lecture </span>
<span class="sd"> Notes in Computer Science, Volume 3418, Springer-Verlag, 2005. </span>
<span class="sd"> http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf</span>
<span class="sd"> </span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">aux_digraph</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">mapping</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">aux_digraph</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">max_flow</span><span class="p">(</span><span class="n">H</span><span class="p">,</span><span class="s">'</span><span class="si">%s</span><span class="s">B'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="s">'</span><span class="si">%s</span><span class="s">A'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
</div>
<div class="viewcode-block" id="node_connectivity"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.node_connectivity.html#networkx.algorithms.connectivity.connectivity.node_connectivity">[docs]</a><span class="k">def</span> <span class="nf">node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns node connectivity for a graph or digraph G.</span>
<span class="sd"> Node connectivity is equal to the minimum number of nodes that </span>
<span class="sd"> must be removed to disconnect G or render it trivial. If source </span>
<span class="sd"> and target nodes are provided, this function returns the local node</span>
<span class="sd"> connectivity: the minimum number of nodes that must be removed to break</span>
<span class="sd"> all paths from source to target in G.</span>
<span class="sd"> This is a flow based implementation. The algorithm is based in </span>
<span class="sd"> solving a number of max-flow problems (ie local st-node connectivity, </span>
<span class="sd"> see local_node_connectivity) to determine the capacity of the</span>
<span class="sd"> minimum cut on an auxiliary directed network that corresponds to the </span>
<span class="sd"> minimum node cut of G. It handles both directed and undirected graphs.</span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional (default=None)</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional (default=None)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> Node connectivity of G, or local node connectivity if source </span>
<span class="sd"> and target were provided</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph is 5-node-connected </span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.node_connectivity(G)</span>
<span class="sd"> 5</span>
<span class="sd"> >>> nx.node_connectivity(G, 3, 7)</span>
<span class="sd"> 5</span>
<span class="sd"> </span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of node connectivity. The </span>
<span class="sd"> algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2)` max-flow </span>
<span class="sd"> problems on an auxiliary digraph. Where `\delta` is the minimum degree </span>
<span class="sd"> of G. For details about the auxiliary digraph and the computation of</span>
<span class="sd"> local node connectivity see local_node_connectivity.</span>
<span class="sd"> This implementation is based on algorithm 11 in [1]_. We use the Ford </span>
<span class="sd"> and Fulkerson algorithm to compute max flow (see ford_fulkerson).</span>
<span class="sd"> </span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> local_node_connectivity</span>
<span class="sd"> all_pairs_node_connectivity_matrix</span>
<span class="sd"> local_edge_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. </span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="c"># Local node connectivity</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">t</span><span class="p">)</span>
<span class="k">return</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="c"># Global node connectivity</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="c"># I think that it is necessary to consider both predecessors</span>
<span class="c"># and successors for directed graphs</span>
<span class="k">def</span> <span class="nf">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
<span class="k">return</span> <span class="n">itertools</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">([</span><span class="n">G</span><span class="o">.</span><span class="n">predecessors_iter</span><span class="p">(</span><span class="n">v</span><span class="p">),</span>
<span class="n">G</span><span class="o">.</span><span class="n">successors_iter</span><span class="p">(</span><span class="n">v</span><span class="p">)])</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">neighbors_iter</span>
<span class="c"># Initial guess \kappa = n - 1</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span><span class="o">-</span><span class="mi">1</span>
<span class="n">deg</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
<span class="n">min_deg</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">deg</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">v</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span><span class="n">d</span> <span class="ow">in</span> <span class="n">deg</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span><span class="o">==</span><span class="n">min_deg</span><span class="p">)</span>
<span class="c"># Reuse the auxiliary digraph</span>
<span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="c"># compute local node connectivity with all non-neighbors nodes</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">]):</span>
<span class="n">K</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span>
<span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">))</span>
<span class="c"># Same for non adjacent pairs of neighbors of v</span>
<span class="k">for</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
<span class="k">if</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">x</span><span class="p">]:</span> <span class="k">continue</span>
<span class="n">K</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span>
<span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">))</span>
<span class="k">return</span> <span class="n">K</span>
</div>
<div class="viewcode-block" id="all_pairs_node_connectivity_matrix"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity_matrix.html#networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity_matrix">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_node_connectivity_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">"""Return a numpy 2d ndarray with node connectivity between all pairs</span>
<span class="sd"> of nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected graph</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : 2d numpy ndarray</span>
<span class="sd"> node connectivity between all pairs of nodes.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> local_node_connectivity</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> local_edge_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> """</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">numpy</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span>\
<span class="s">"all_pairs_node_connectivity_matrix() requires NumPy"</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="c"># Create auxiliary Digraph</span>
<span class="n">D</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">D</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">)</span>
<span class="n">M</span><span class="p">[</span><span class="n">mapping</span><span class="p">[</span><span class="n">u</span><span class="p">],</span><span class="n">mapping</span><span class="p">[</span><span class="n">v</span><span class="p">]]</span> <span class="o">=</span> <span class="n">K</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">K</span> <span class="o">=</span> <span class="n">local_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">D</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">)</span>
<span class="n">M</span><span class="p">[</span><span class="n">mapping</span><span class="p">[</span><span class="n">u</span><span class="p">],</span><span class="n">mapping</span><span class="p">[</span><span class="n">v</span><span class="p">]]</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="n">mapping</span><span class="p">[</span><span class="n">v</span><span class="p">],</span><span class="n">mapping</span><span class="p">[</span><span class="n">u</span><span class="p">]]</span> <span class="o">=</span> <span class="n">K</span>
<span class="k">return</span> <span class="n">M</span>
</div>
<span class="k">def</span> <span class="nf">_aux_digraph_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">"""Auxiliary digraph for computing flow based edge connectivity</span>
<span class="sd"> </span>
<span class="sd"> If the input graph is undirected, we replace each edge (u,v) with</span>
<span class="sd"> two reciprocal arcs (u,v) and (v,u) and then we set the attribute </span>
<span class="sd"> 'capacity' for each arc to 1. If the input graph is directed we simply</span>
<span class="sd"> add the 'capacity' attribute. Part of algorithm 1 in [1]_ .</span>
<span class="sd"> </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. (this is a </span>
<span class="sd"> chapter, look for the reference of the book).</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">if</span> <span class="n">nx</span><span class="o">.</span><span class="n">get_edge_attributes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="s">'capacity'</span><span class="p">):</span>
<span class="k">return</span> <span class="n">G</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">capacity</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">e</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">D</span><span class="o">.</span><span class="n">edges</span><span class="p">())</span>
<span class="n">nx</span><span class="o">.</span><span class="n">set_edge_attributes</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="s">'capacity'</span><span class="p">,</span> <span class="n">capacity</span><span class="p">)</span>
<span class="k">return</span> <span class="n">D</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">to_directed</span><span class="p">()</span>
<span class="n">capacity</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">e</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">D</span><span class="o">.</span><span class="n">edges</span><span class="p">())</span>
<span class="n">nx</span><span class="o">.</span><span class="n">set_edge_attributes</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="s">'capacity'</span><span class="p">,</span> <span class="n">capacity</span><span class="p">)</span>
<span class="k">return</span> <span class="n">D</span>
<div class="viewcode-block" id="local_edge_connectivity"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html#networkx.algorithms.connectivity.connectivity.local_edge_connectivity">[docs]</a><span class="k">def</span> <span class="nf">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns local edge connectivity for nodes s and t in G.</span>
<span class="sd"> Local edge connectivity for two nodes s and t is the minimum number </span>
<span class="sd"> of edges that must be removed to disconnect them. </span>
<span class="sd"> This is a flow based implementation of edge connectivity. We compute the</span>
<span class="sd"> maximum flow on an auxiliary digraph build from the original</span>
<span class="sd"> network (see below for details). This is equal to the local edge </span>
<span class="sd"> connectivity because the value of a maximum s-t-flow is equal to the </span>
<span class="sd"> capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected or directed graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node</span>
<span class="sd"> aux_digraph : NetworkX DiGraph (default=None)</span>
<span class="sd"> Auxiliary digraph to compute flow based edge connectivity. If None</span>
<span class="sd"> the auxiliary digraph is build.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> local edge connectivity for nodes s and t</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph has edge connectivity 5 </span>
<span class="sd"> >>> # for each non adjacent node pair</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.local_edge_connectivity(G,0,6)</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of edge connectivity. We compute the</span>
<span class="sd"> maximum flow using the Ford and Fulkerson algorithm on an auxiliary digraph </span>
<span class="sd"> build from the original graph:</span>
<span class="sd"> If the input graph is undirected, we replace each edge (u,v) with</span>
<span class="sd"> two reciprocal arcs `(u,v)` and `(v,u)` and then we set the attribute </span>
<span class="sd"> 'capacity' for each arc to 1. If the input graph is directed we simply</span>
<span class="sd"> add the 'capacity' attribute. This is an implementation of algorithm 1 </span>
<span class="sd"> in [1]_.</span>
<span class="sd"> </span>
<span class="sd"> The maximum flow in the auxiliary network is equal to the local edge </span>
<span class="sd"> connectivity because the value of a maximum s-t-flow is equal to the </span>
<span class="sd"> capacity of a minimum s-t-cut (Ford and Fulkerson theorem).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> local_node_connectivity</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> </span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">aux_digraph</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">_aux_digraph_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">aux_digraph</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">max_flow</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
</div>
<div class="viewcode-block" id="edge_connectivity"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.connectivity.edge_connectivity.html#networkx.algorithms.connectivity.connectivity.edge_connectivity">[docs]</a><span class="k">def</span> <span class="nf">edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns the edge connectivity of the graph or digraph G.</span>
<span class="sd"> The edge connectivity is equal to the minimum number of edges that </span>
<span class="sd"> must be removed to disconnect G or render it trivial. If source </span>
<span class="sd"> and target nodes are provided, this function returns the local edge</span>
<span class="sd"> connectivity: the minimum number of edges that must be removed to </span>
<span class="sd"> break all paths from source to target in G.</span>
<span class="sd"> </span>
<span class="sd"> This is a flow based implementation. The algorithm is based in solving </span>
<span class="sd"> a number of max-flow problems (ie local st-edge connectivity, see </span>
<span class="sd"> local_edge_connectivity) to determine the capacity of the minimum </span>
<span class="sd"> cut on an auxiliary directed network that corresponds to the minimum </span>
<span class="sd"> edge cut of G. It handles both directed and undirected graphs.</span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Undirected or directed graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional (default=None)</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional (default=None)</span>
<span class="sd"> </span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> K : integer</span>
<span class="sd"> Edge connectivity for G, or local edge connectivity if source </span>
<span class="sd"> and target were provided</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph is 5-edge-connected</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> nx.edge_connectivity(G)</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of global edge connectivity. For</span>
<span class="sd"> undirected graphs the algorithm works by finding a 'small' dominating </span>
<span class="sd"> set of nodes of G (see algorithm 7 in [1]_ ) and computing local max flow </span>
<span class="sd"> (see local_edge_connectivity) between an arbitrary node in the dominating </span>
<span class="sd"> set and the rest of nodes in it. This is an implementation of </span>
<span class="sd"> algorithm 6 in [1]_ .</span>
<span class="sd"> For directed graphs, the algorithm does n calls to the max flow function.</span>
<span class="sd"> This is an implementation of algorithm 8 in [1]_ . We use the Ford and </span>
<span class="sd"> Fulkerson algorithm to compute max flow (see ford_fulkerson).</span>
<span class="sd"> </span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> local_node_connectivity</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> local_edge_connectivity</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. </span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="c"># Local edge connectivity</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">t</span><span class="p">)</span>
<span class="k">return</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="c"># Global edge connectivity</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="c"># Algorithm 8 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="c"># initial value for lambda is min degree (\delta(G))</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="c"># reuse auxiliary digraph</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">_aux_digraph_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">()</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodes</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
<span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">))</span>
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span> <span class="c"># last node!</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
<span class="n">nodes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">))</span>
<span class="k">return</span> <span class="n">L</span>
<span class="k">else</span><span class="p">:</span> <span class="c"># undirected</span>
<span class="c"># Algorithm 6 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="c"># initial value for lambda is min degree (\delta(G))</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="c"># reuse auxiliary digraph</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">_aux_digraph_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="c"># A dominating set is \lambda-covering</span>
<span class="c"># We need a dominating set with at least two nodes</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">dominating_set</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">start_with</span><span class="o">=</span><span class="n">node</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">D</span><span class="p">:</span> <span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="c"># in complete graphs the dominating sets will always be of one node</span>
<span class="c"># thus we return min degree</span>
<span class="k">return</span> <span class="n">L</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">D</span><span class="p">:</span>
<span class="n">L</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">local_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">))</span>
<span class="k">return</span> <span class="n">L</span>
</div>
<span class="k">def</span> <span class="nf">dominating_set</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">start_with</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="c"># Algorithm 7 in [1]</span>
<span class="n">all_nodes</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">if</span> <span class="n">start_with</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">v</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span> <span class="c"># pick a node</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="n">start_with</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in G'</span> <span class="o">%</span> <span class="n">start_with</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">start_with</span>
<span class="n">D</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">])</span>
<span class="n">ND</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">nbr</span> <span class="k">for</span> <span class="n">nbr</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">]])</span>
<span class="n">other</span> <span class="o">=</span> <span class="n">all_nodes</span> <span class="o">-</span> <span class="n">ND</span> <span class="o">-</span> <span class="n">D</span>
<span class="k">while</span> <span class="n">other</span><span class="p">:</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">D</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">ND</span><span class="o">.</span><span class="n">update</span><span class="p">([</span><span class="n">nbr</span> <span class="k">for</span> <span class="n">nbr</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">w</span><span class="p">]</span> <span class="k">if</span> <span class="n">nbr</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">D</span><span class="p">])</span>
<span class="n">other</span> <span class="o">=</span> <span class="n">all_nodes</span> <span class="o">-</span> <span class="n">ND</span> <span class="o">-</span> <span class="n">D</span>
<span class="k">return</span> <span class="n">D</span>
<span class="k">def</span> <span class="nf">is_dominating_set</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nbunch</span><span class="p">):</span>
<span class="c"># Proposed by Dan on the mailing list</span>
<span class="n">allnodes</span><span class="o">=</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">testset</span><span class="o">=</span><span class="nb">set</span><span class="p">(</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nbunch</span> <span class="k">if</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">)</span>
<span class="n">nbrs</span><span class="o">=</span><span class="nb">set</span><span class="p">()</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">testset</span><span class="p">:</span>
<span class="n">nbrs</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="k">if</span> <span class="n">nbrs</span> <span class="o">-</span> <span class="n">allnodes</span><span class="p">:</span> <span class="c"># some nodes left--not dominating</span>
<span class="k">return</span> <span class="bp">False</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">True</span>
</pre></div>
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