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<div class="section" id="articulation-points">
<h1>articulation_points<a class="headerlink" href="#articulation-points" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.components.biconnected.articulation_points">
<tt class="descname">articulation_points</tt><big>(</big><em>G</em><big>)</big><a class="reference internal" href="../../_modules/networkx/algorithms/components/biconnected.html#articulation_points"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.algorithms.components.biconnected.articulation_points" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a generator of articulation points, or cut vertices, of a graph.</p>
<p>An articulation point or cut vertex is any node whose removal (along with
all its incident edges) increases the number of connected components of
a graph. An undirected connected graph without articulation points is
biconnected. Articulation points belong to more than one biconnected
component of a graph.</p>
<p>Notice that by convention a dyad is considered a biconnected component.</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>An undirected graph.</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>articulation points</strong> : generator</p>
<blockquote>
<div><p>generator of nodes</p>
</div></blockquote>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NetworkXError :</strong> :</p>
<blockquote class="last">
<div><p>If the input graph is not undirected.</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.components.biconnected.is_biconnected.html#networkx.algorithms.components.biconnected.is_biconnected" title="networkx.algorithms.components.biconnected.is_biconnected"><tt class="xref py py-obj docutils literal"><span class="pre">is_biconnected</span></tt></a>, <a class="reference internal" href="networkx.algorithms.components.biconnected.biconnected_components.html#networkx.algorithms.components.biconnected.biconnected_components" title="networkx.algorithms.components.biconnected.biconnected_components"><tt class="xref py py-obj docutils literal"><span class="pre">biconnected_components</span></tt></a>, <a class="reference internal" href="networkx.algorithms.components.biconnected.biconnected_component_edges.html#networkx.algorithms.components.biconnected.biconnected_component_edges" title="networkx.algorithms.components.biconnected.biconnected_component_edges"><tt class="xref py py-obj docutils literal"><span class="pre">biconnected_component_edges</span></tt></a>, <a class="reference internal" href="networkx.algorithms.components.biconnected.biconnected_component_subgraphs.html#networkx.algorithms.components.biconnected.biconnected_component_subgraphs" title="networkx.algorithms.components.biconnected.biconnected_component_subgraphs"><tt class="xref py py-obj docutils literal"><span class="pre">biconnected_component_subgraphs</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>The algorithm to find articulation points and biconnected
components is implemented using a non-recursive depth-first-search
(DFS) that keeps track of the highest level that back edges reach
in the DFS tree. A node <span class="math">n</span> is an articulation point if, and only
if, there exists a subtree rooted at <span class="math">n</span> such that there is no
back edge from any successor of <span class="math">n</span> that links to a predecessor of
<span class="math">n</span> in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all
edges of a bicomponent will be traversed consecutively between
articulation points.</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r188" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[R188]</a></td><td>Hopcroft, J.; Tarjan, R. (1973).
“Efficient algorithms for graph manipulation”.
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272</td></tr>
</tbody>
</table>
<p class="rubric">Examples</p>
<div class="highlight-python"><div class="highlight"><pre><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">barbell_graph</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">is_biconnected</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">articulation_points</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="go">[6, 5, 4, 3]</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">is_biconnected</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">articulation_points</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="go">[]</span>
</pre></div>
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