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<div class="section" id="is-valid-degree-sequence-erdos-gallai">
<h1>is_valid_degree_sequence_erdos_gallai<a class="headerlink" href="#is-valid-degree-sequence-erdos-gallai" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.graphical.is_valid_degree_sequence_erdos_gallai">
<tt class="descname">is_valid_degree_sequence_erdos_gallai</tt><big>(</big><em>deg_sequence</em><big>)</big><a class="reference internal" href="../../_modules/networkx/algorithms/graphical.html#is_valid_degree_sequence_erdos_gallai"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#networkx.algorithms.graphical.is_valid_degree_sequence_erdos_gallai" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns True if deg_sequence can be realized by a simple graph.</p>
<p>The validation is done using the Erdős-Gallai theorem <a class="reference internal" href="../../bibliography.html#eg1960">[EG1960]</a>.</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>deg_sequence</strong> : list</p>
<blockquote>
<div><p>A list of integers</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>valid</strong> : bool</p>
<blockquote class="last">
<div><p>True if deg_sequence is graphical and False if not.</p>
</div></blockquote>
</td>
</tr>
</tbody>
</table>
<p class="rubric">Notes</p>
<p>This implementation uses an equivalent form of the Erdős-Gallai criterion.
Worst-case run time is: O(n) where n is the length of the sequence.</p>
<p>Specifically, a sequence d is graphical if and only if the
sum of the sequence is even and for all strong indices k in the sequence,</p>
<blockquote>
<div><div class="math">
<p><span class="math">\sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k)
= k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j )</span></p>
</div></div></blockquote>
<p>A strong index k is any index where <span class="math">d_k \geq k</span> and the value <span class="math">n_j</span> is the
number of occurrences of j in d. The maximal strong index is called the
Durfee index.</p>
<p>This particular rearrangement comes from the proof of Theorem 3 in <a class="reference internal" href="#r226">[R226]</a>.</p>
<p>The ZZ condition says that for the sequence d if</p>
<div class="math">
<p><span class="math">|d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)}</span></p>
</div><p>then d is graphical. This was shown in Theorem 6 in <a class="reference internal" href="#r226">[R226]</a>.</p>
<p class="rubric">References</p>
<table class="docutils citation" frame="void" id="r225" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id6">[R225]</a></td><td>A. Tripathi and S. Vijay. “A note on a theorem of Erdős & Gallai”,
Discrete Mathematics, 265, pp. 417-420 (2003).</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="r226" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[R226]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id3">2</a>, <a class="fn-backref" href="#id7">3</a>)</em> I.E. Zverovich and V.E. Zverovich. “Contributions to the theory
of graphic sequences”, Discrete Mathematics, 105, pp. 292-303 (1992).</td></tr>
</tbody>
</table>
<p><a class="reference internal" href="../../bibliography.html#eg1960">[EG1960]</a>, <a class="reference internal" href="../../bibliography.html#choudum1986">[choudum1986]</a></p>
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