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  <div class="section" id="node-connectivity">
<h1>node_connectivity<a class="headerlink" href="#node-connectivity" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.connectivity.connectivity.node_connectivity">
<tt class="descname">node_connectivity</tt><big>(</big><em>G</em>, <em>s=None</em>, <em>t=None</em><big>)</big><a class="reference internal" href="../../_modules/networkx/algorithms/connectivity/connectivity.html#node_connectivity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.algorithms.connectivity.connectivity.node_connectivity" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns node connectivity for a graph or digraph G.</p>
<p>Node connectivity is equal to the minimum number of nodes that 
must be removed to disconnect G or render it trivial. If source 
and target nodes are provided, this function returns the local node
connectivity: the minimum number of nodes that must be removed to break
all paths from source to target in G.</p>
<p>This is a flow based implementation. 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.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>G</strong> : NetworkX graph</p>
<blockquote>
<div><p>Undirected graph</p>
</div></blockquote>
<p><strong>s</strong> : node</p>
<blockquote>
<div><p>Source node. Optional (default=None)</p>
</div></blockquote>
<p><strong>t</strong> : node</p>
<blockquote>
<div><p>Target node. Optional (default=None)</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>K</strong> : integer</p>
<blockquote class="last">
<div><p>Node connectivity of G, or local node connectivity if source 
and target were provided</p>
</div></blockquote>
</td>
</tr>
</tbody>
</table>
<div class="admonition-see-also admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="networkx.algorithms.connectivity.connectivity.local_node_connectivity.html#networkx.algorithms.connectivity.connectivity.local_node_connectivity" title="networkx.algorithms.connectivity.connectivity.local_node_connectivity"><tt class="xref py py-obj docutils literal"><span class="pre">local_node_connectivity</span></tt></a>, <a class="reference internal" href="networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity_matrix.html#networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity_matrix" title="networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">all_pairs_node_connectivity_matrix</span></tt></a>, <a class="reference internal" href="networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html#networkx.algorithms.connectivity.connectivity.local_edge_connectivity" title="networkx.algorithms.connectivity.connectivity.local_edge_connectivity"><tt class="xref py py-obj docutils literal"><span class="pre">local_edge_connectivity</span></tt></a>, <a class="reference internal" href="networkx.algorithms.connectivity.connectivity.edge_connectivity.html#networkx.algorithms.connectivity.connectivity.edge_connectivity" title="networkx.algorithms.connectivity.connectivity.edge_connectivity"><tt class="xref py py-obj docutils literal"><span class="pre">edge_connectivity</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">max_flow</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">ford_fulkerson</span></tt></p>
</div>
<p class="rubric">Notes</p>
<p>This is a flow based implementation of node connectivity. The 
algorithm works by solving <span class="math">O((n-\delta-1+\delta(\delta-1)/2)</span> max-flow 
problems on an auxiliary digraph. Where <span class="math">\delta</span> is the minimum degree 
of G. For details about the auxiliary digraph and the computation of
local node connectivity see local_node_connectivity.</p>
<p>This implementation is based on algorithm 11 in <a class="reference internal" href="#r202">[R202]</a>. We use the Ford 
and Fulkerson algorithm to compute max flow (see ford_fulkerson).</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r202" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R202]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> Abdol-Hossein Esfahanian. Connectivity Algorithms. 
<a class="reference external" href="http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf">http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</a></td></tr>
</tbody>
</table>
<p class="rubric">Examples</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># Platonic icosahedral graph is 5-node-connected </span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">icosahedral_graph</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">nx</span><span class="o">.</span><span class="n">node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="go">5</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">nx</span><span class="o">.</span><span class="n">node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="go">5</span>
</pre></div>
</div>
</dd></dl>

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