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<head><title>independenceNumber -- determines the independence number of a graph</title>
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<div><h1>independenceNumber -- determines the independence number of a graph</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>d = independenceNumber G</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>a <a href="___Graph.html">graph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>d</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the independence number of <tt>G</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function returns the maximum number of independent vertices in a graph. This number can be found by computing the dimension of the simplicial complex whose faces are the independent sets (see <a href="_independence__Complex.html" title="returns the independence complex of a (hyper)graph">independenceComplex</a>) and adding 1 to this number.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..e];</pre>
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<tr><td><pre>i2 : c4 = graph {a*b,b*c,c*d,d*a} -- 4-cycle plus an isolated vertex!!!!
o2 = Graph{edges => {{a, b}, {b, c}, {a, d}, {c, d}}}
ring => R
vertices => {a, b, c, d, e}
o2 : Graph</pre>
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<tr><td><pre>i3 : c5 = graph {a*b,b*c,c*d,d*e,e*a} -- 5-cycle
o3 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o3 : Graph</pre>
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<tr><td><pre>i4 : independenceNumber c4
o4 = 3</pre>
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<tr><td><pre>i5 : independenceNumber c5
o5 = 2</pre>
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<tr><td><pre>i6 : dim independenceComplex c4 + 1 == independenceNumber c4
o6 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_independence__Complex.html" title="returns the independence complex of a (hyper)graph">independenceComplex</a> -- returns the independence complex of a (hyper)graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>independenceNumber</tt> :</h2>
<ul><li>independenceNumber(Graph)</li>
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