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<head><title>independenceComplex -- returns the independence complex of a (hyper)graph</title>
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<div><h1>independenceComplex -- returns the independence complex of a (hyper)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 = independenceComplex H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>D</tt>, <span>a <a href="../../SimplicialComplexes/html/___Simplicial__Complex.html">simplicial complex</a></span>, the independence complex associated to the (hyper)graph</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function associates to a (hyper)graph a simplicial complex whose faces correspond to the independent sets of the (hyper)graph. See, for example, the paper of A. Van Tuyl and R. Villarreal "Shellable graphs and sequentially Cohen-Macaulay bipartite graphs," Journal of Combinatorial Theory, Series A 115 (2008) 799-814.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a..e];</pre>
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<tr><td><pre>i2 : g = graph {a*b,b*c,c*d,d*e,e*a} -- the 5-cycle
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o2 : Graph</pre>
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<tr><td><pre>i3 : independenceComplex g
o3 = | ce be bd ad ac |
o3 : SimplicialComplex</pre>
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<tr><td><pre>i4 : h = hyperGraph {a*b*c,b*c*d,d*e}
o4 = HyperGraph{edges => {{a, b, c}, {b, c, d}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o4 : HyperGraph</pre>
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<tr><td><pre>i5 : independenceComplex h
o5 = | bce ace abe acd abd |
o5 : SimplicialComplex</pre>
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<div>Equivalently, the independence complex is the simplicial complex associated to the edge ideal of the (hyper)graph <tt>H</tt> via the Stanley-Reisner correspondence.</div>
<table class="examples"><tr><td><pre>i6 : S = QQ[a..e];</pre>
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<tr><td><pre>i7 : g = graph {a*b,b*c,a*c,d*e,a*e}
o7 = Graph{edges => {{a, b}, {a, c}, {b, c}, {a, e}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o7 : Graph</pre>
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<tr><td><pre>i8 : Delta1 = independenceComplex g
o8 = | ce be cd bd ad |
o8 : SimplicialComplex</pre>
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<tr><td><pre>i9 : Delta2 = simplicialComplex edgeIdeal g
o9 = | ce be cd bd ad |
o9 : SimplicialComplex</pre>
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<tr><td><pre>i10 : Delta1 == Delta2
o10 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_independence__Number.html" title="determines the independence number of a graph">independenceNumber</a> -- determines the independence number of a graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>independenceComplex</tt> :</h2>
<ul><li>independenceComplex(HyperGraph)</li>
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