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<head><title>coverIdeal -- creates the cover ideal of a (hyper)graph</title>
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<div><h1>coverIdeal -- creates the cover ideal 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>I = coverIdeal 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>I</tt>, <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, the cover ideal of H</span></li>
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<div class="single"><h2>Description</h2>
<div><div>Returns the monomial ideal generated by the minimal vertex covers. This is also the Alexander dual of the edge ideal of the hypergraph <tt>H</tt>.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a,b,c,d,e,f];</pre>
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<tr><td><pre>i2 : k6 = completeGraph S -- complete graph on 6 vertices
o2 = Graph{edges => {{a, b}, {a, c}, {a, d}, {a, e}, {a, f}, {b, c}, {b, d}, {b, e}, {b, f}, {c, d}, {c, e}, {c, f}, {d, e}, {d, f}, {e, f}}}
ring => S
vertices => {a, b, c, d, e, f}
o2 : Graph</pre>
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<tr><td><pre>i3 : coverIdeal k6 -- each generator corresponds to a minimal vertex of k6
o3 = monomialIdeal (a*b*c*d*e, a*b*c*d*f, a*b*c*e*f, a*b*d*e*f, a*c*d*e*f,
------------------------------------------------------------------------
b*c*d*e*f)
o3 : MonomialIdeal of S</pre>
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<tr><td><pre>i4 : h = hyperGraph {a*b*c,c*d,d*e*f}
o4 = HyperGraph{edges => {{a, b, c}, {c, d}, {d, e, f}}}
ring => S
vertices => {a, b, c, d, e, f}
o4 : HyperGraph</pre>
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<tr><td><pre>i5 : coverIdeal h
o5 = monomialIdeal (a*d, b*d, c*d, c*e, c*f)
o5 : MonomialIdeal of S</pre>
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<tr><td><pre>i6 : dual coverIdeal h == edgeIdeal h
o6 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../Macaulay2Doc/html/_dual.html" title="dual module or map">dual</a> -- dual module or map</span></li>
<li><span><a href="_edge__Ideal.html" title="creates the edge ideal of a (hyper)graph">edgeIdeal</a> -- creates the edge ideal of a (hyper)graph</span></li>
<li><span><a href="_vertex__Cover__Number.html" title="find the vertex covering number of a (hyper)graph">vertexCoverNumber</a> -- find the vertex covering number of a (hyper)graph</span></li>
<li><span><a href="_vertex__Covers.html" title="list the minimal vertex covers of a (hyper)graph">vertexCovers</a> -- list the minimal vertex covers of a (hyper)graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>coverIdeal</tt> :</h2>
<ul><li>coverIdeal(HyperGraph)</li>
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