<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>node_connectivity — NetworkX 1.8.1 documentation</title> <link rel="stylesheet" href="../../_static/networkx.css" type="text/css" /> <link rel="stylesheet" href="../../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../../', VERSION: '1.8.1', COLLAPSE_INDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: false }; </script> <script type="text/javascript" src="../../_static/jquery.js"></script> <script type="text/javascript" src="../../_static/underscore.js"></script> <script type="text/javascript" src="../../_static/doctools.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within NetworkX 1.8.1 documentation" href="../../_static/opensearch.xml"/> <link rel="top" title="NetworkX 1.8.1 documentation" href="../../index.html" /> <link rel="up" title="Connectivity" href="../algorithms.connectivity.html" /> <link rel="next" title="local_edge_connectivity" href="networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html" /> <link rel="prev" title="local_node_connectivity" href="networkx.algorithms.connectivity.connectivity.local_node_connectivity.html" /> </head> <body> <div style="color: black;background-color: white; font-size: 3.2em; text-align: left; padding: 15px 10px 10px 15px"> NetworkX </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html" title="local_edge_connectivity" accesskey="N">next</a> |</li> <li class="right" > <a href="networkx.algorithms.connectivity.connectivity.local_node_connectivity.html" title="local_node_connectivity" accesskey="P">previous</a> |</li> <li><a href="http://networkx.github.com/">NetworkX Home </a> | </li> <li><a href="http://networkx.github.com/documentation.html">Documentation </a>| </li> <li><a href="http://networkx.github.com/download.html">Download </a> | </li> <li><a href="http://github.com/networkx">Developer (Github)</a></li> <li><a href="../index.html" >Reference</a> »</li> <li><a href="../pdf_reference.html" >Reference</a> »</li> <li><a href="../algorithms.html" >Algorithms</a> »</li> <li><a href="../algorithms.connectivity.html" accesskey="U">Connectivity</a> »</li> </ul> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="networkx.algorithms.connectivity.connectivity.local_node_connectivity.html" title="previous chapter">local_node_connectivity</a></p> <h4>Next topic</h4> <p class="topless"><a href="networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html" title="next chapter">local_edge_connectivity</a></p> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> <p class="searchtip" style="font-size: 90%"> Enter search terms or a module, class or function name. </p> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <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">>>> </span><span class="c"># Platonic icosahedral graph is 5-node-connected </span> <span class="gp">>>> </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">>>> </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">>>> </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> </div> </div> </div> </div> <div class="clearer"></div> </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="networkx.algorithms.connectivity.connectivity.local_edge_connectivity.html" title="local_edge_connectivity" >next</a> |</li> <li class="right" > <a href="networkx.algorithms.connectivity.connectivity.local_node_connectivity.html" title="local_node_connectivity" >previous</a> |</li> <li><a href="http://networkx.github.com/">NetworkX Home </a> | </li> <li><a href="http://networkx.github.com/documentation.html">Documentation </a>| </li> <li><a href="http://networkx.github.com/download.html">Download </a> | </li> <li><a href="http://github.com/networkx">Developer (Github)</a></li> <li><a href="../index.html" >Reference</a> »</li> <li><a href="../pdf_reference.html" >Reference</a> »</li> <li><a href="../algorithms.html" >Algorithms</a> »</li> <li><a href="../algorithms.connectivity.html" >Connectivity</a> »</li> </ul> </div> <div class="footer"> © Copyright 2013, NetworkX Developers. 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