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<head><title>hyperGraph -- constructor for HyperGraph</title>
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<div><h1>hyperGraph -- constructor for HyperGraph</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>H = hyperGraph(R,E)</tt><br/><tt>H = hyperGraph(I)</tt><br/><tt>H = hyperGraph(J)</tt><br/><tt>H = hyperGraph(E)</tt><br/><tt>H = hyperGraph(G)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, whose variables correspond to vertices of the hypergraph</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of edges, which themselves are lists of vertices</span></li>
<li><span><tt>I</tt>, <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, which must be square-free and whose generators become the edges of the hypergraph</span></li>
<li><span><tt>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which must be square-free monomial and whose generators become the edges of the hypergraph</span></li>
<li><span><tt>G</tt>, <span>a <a href="___Graph.html">graph</a></span>, which is to be converted to a HyperGraph</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</a></span></span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p>The function <tt>hyperGraph</tt> is a constructor for <a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a>. The user can input a hypergraph in a number of different ways, which we describe below. The information decribing the hypergraph is stored in a hash table. We require that there be no inclusion relations between the edges of a hypergraph; that is, that it be a clutter. The reason is that this package is designed for edge ideals, which would lose any information about edges that are supersets of other edges.</p>
<div>For the first possiblity, the user inputs a polynomial ring, which specifices the vertices of graph, and a list of the edges of the graph. The edges are represented as lists.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : E = {{a,b,c},{b,c,d},{c,d,e},{e,d,f}}
o2 = {{a, b, c}, {b, c, d}, {c, d, e}, {e, d, f}}
o2 : List</pre>
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<tr><td><pre>i3 : h = hyperGraph (R,E)
o3 = HyperGraph{edges => {{a, b, c}, {b, c, d}, {c, d, e}, {e, d, f}}}
ring => R
vertices => {a, b, c, d, e, f}
o3 : HyperGraph</pre>
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<div>Alternatively, if the polynomial ring has already been defined, it suffices simply to enter the list of the edges.</div>
<table class="examples"><tr><td><pre>i4 : S = QQ[z_1..z_8]
o4 = S
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : E1 = {{z_1,z_2,z_3},{z_2,z_4,z_5,z_6},{z_4,z_7,z_8},{z_5,z_7,z_8}}
o5 = {{z , z , z }, {z , z , z , z }, {z , z , z }, {z , z , z }}
1 2 3 2 4 5 6 4 7 8 5 7 8
o5 : List</pre>
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<tr><td><pre>i6 : E2 = {{z_2,z_3,z_4},{z_4,z_5}}
o6 = {{z , z , z }, {z , z }}
2 3 4 4 5
o6 : List</pre>
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<tr><td><pre>i7 : h1 = hyperGraph E1
o7 = HyperGraph{edges => {{z , z , z }, {z , z , z , z }, {z , z , z }, {z , z , z }}}
1 2 3 2 4 5 6 4 7 8 5 7 8
ring => S
vertices => {z , z , z , z , z , z , z , z }
1 2 3 4 5 6 7 8
o7 : HyperGraph</pre>
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<tr><td><pre>i8 : h2 = hyperGraph E2
o8 = HyperGraph{edges => {{z , z , z }, {z , z }} }
2 3 4 4 5
ring => S
vertices => {z , z , z , z , z , z , z , z }
1 2 3 4 5 6 7 8
o8 : HyperGraph</pre>
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<div>The list of edges could also be entered as a list of square-free monomials.</div>
<table class="examples"><tr><td><pre>i9 : T = QQ[w,x,y,z]
o9 = T
o9 : PolynomialRing</pre>
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<tr><td><pre>i10 : e = {w*x*y,w*x*z,w*y*z,x*y*z}
o10 = {w*x*y, w*x*z, w*y*z, x*y*z}
o10 : List</pre>
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<tr><td><pre>i11 : h = hyperGraph e
o11 = HyperGraph{edges => {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}}
ring => T
vertices => {w, x, y, z}
o11 : HyperGraph</pre>
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<div>Another option for defining an hypergraph is to use an <a href="../../Macaulay2Doc/html/_ideal.html" title="make an ideal">ideal</a> or <a href="../../Macaulay2Doc/html/_monomial__Ideal.html" title="make a monomial ideal">monomialIdeal</a>.</div>
<table class="examples"><tr><td><pre>i12 : C = QQ[p_1..p_6]
o12 = C
o12 : PolynomialRing</pre>
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<tr><td><pre>i13 : i = monomialIdeal (p_1*p_2*p_3,p_3*p_4*p_5,p_3*p_6)
o13 = monomialIdeal (p p p , p p p , p p )
1 2 3 3 4 5 3 6
o13 : MonomialIdeal of C</pre>
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<tr><td><pre>i14 : hyperGraph i
o14 = HyperGraph{edges => {{p , p , p }, {p , p , p }, {p , p }}}
1 2 3 3 4 5 3 6
ring => C
vertices => {p , p , p , p , p , p }
1 2 3 4 5 6
o14 : HyperGraph</pre>
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<tr><td><pre>i15 : j = ideal (p_1*p_2,p_3*p_4*p_5,p_6)
o15 = ideal (p p , p p p , p )
1 2 3 4 5 6
o15 : Ideal of C</pre>
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<tr><td><pre>i16 : hyperGraph j
o16 = HyperGraph{edges => {{p , p }, {p , p , p }, {p }}}
1 2 3 4 5 6
ring => C
vertices => {p , p , p , p , p , p }
1 2 3 4 5 6
o16 : HyperGraph</pre>
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<div>From any graph we can make a hypergraph with the same edges.</div>
<table class="examples"><tr><td><pre>i17 : D = QQ[r_1..r_5]
o17 = D
o17 : PolynomialRing</pre>
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<tr><td><pre>i18 : g = graph {r_1*r_2,r_2*r_4,r_3*r_5,r_5*r_4,r_1*r_5}
o18 = Graph{edges => {{r , r }, {r , r }, {r , r }, {r , r }, {r , r }}}
1 2 2 4 1 5 3 5 4 5
ring => D
vertices => {r , r , r , r , r }
1 2 3 4 5
o18 : Graph</pre>
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<tr><td><pre>i19 : h = hyperGraph g
o19 = HyperGraph{edges => {{r , r }, {r , r }, {r , r }, {r , r }, {r , r }}}
1 2 2 4 1 5 3 5 4 5
ring => D
vertices => {r , r , r , r , r }
1 2 3 4 5
o19 : HyperGraph</pre>
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<div>Not all hypergraph constructors are able to make the empty hypergraph, that is, the hypergraph with no edges. Specifically, the constructors that take only a list cannot make the empty hypergraph because the underlying ring is not given. To define the empty hypergraph, give an explicit polynomial ring or give the (monomial) ideal.</div>
<table class="examples"><tr><td><pre>i20 : E = QQ[m,n,o,p]
o20 = E
o20 : PolynomialRing</pre>
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<tr><td><pre>i21 : hyperGraph(E, {})
o21 = HyperGraph{edges => {} }
ring => E
vertices => {m, n, o, p}
o21 : HyperGraph</pre>
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<tr><td><pre>i22 : hyperGraph monomialIdeal(0_E) -- the zero element of E (do not use 0)
o22 = HyperGraph{edges => {} }
ring => E
vertices => {m, n, o, p}
o22 : HyperGraph</pre>
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<tr><td><pre>i23 : hyperGraph ideal (0_E)
o23 = HyperGraph{edges => {} }
ring => E
vertices => {m, n, o, p}
o23 : HyperGraph</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></li>
<li><span><a href="___Constructor_sp__Overview.html" title="a summary of the many ways of making graphs and hypergraphs">Constructor Overview</a> -- a summary of the many ways of making graphs and hypergraphs</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>hyperGraph</tt> :</h2>
<ul><li>hyperGraph(Graph)</li>
<li>hyperGraph(Ideal)</li>
<li>hyperGraph(List)</li>
<li>hyperGraph(MonomialIdeal)</li>
<li>hyperGraph(PolynomialRing,List)</li>
</ul>
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