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<head><title>allOddHoles -- returns all odd holes in a graph</title>
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<div><h1>allOddHoles -- returns all odd holes in 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>L = allOddHoles 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>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, returns all odd holes contained in <tt>G</tt>.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>An odd hole is an odd induced cycle of length at least 5. The method is based on work of Francisco-Ha-Van Tuyl, looking at the associated primes of the square of the Alexander dual of the edge ideal.</p>
<div>See C.A. Francisco, H.T. Ha, A. Van Tuyl, "Algebraic methods for detecting odd holes in a graph." (2008) Preprint. <tt>arXiv:0806.1159v1</tt>.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x_1..x_6];</pre>
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<tr><td><pre>i2 : G = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6}) --5-cycle and a triangle
o2 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 3 4 1 5 4 5 1 6 5 6
ring => R
vertices => {x , x , x , x , x , x }
1 2 3 4 5 6
o2 : Graph</pre>
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<tr><td><pre>i3 : allOddHoles G --only the 5-cycle should appear
o3 = {{x , x , x , x , x }}
1 2 3 4 5
o3 : List</pre>
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<tr><td><pre>i4 : H = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6,x_1*x_4}) --no odd holes
o4 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 1 4 3 4 1 5 4 5 1 6 5 6
ring => R
vertices => {x , x , x , x , x , x }
1 2 3 4 5 6
o4 : Graph</pre>
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<tr><td><pre>i5 : allOddHoles H
o5 = {}
o5 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_all__Even__Holes.html" title="returns all even holes in a graph">allEvenHoles</a> -- returns all even holes in a graph</span></li>
<li><span><a href="_has__Odd__Hole.html" title="tells whether a graph contains an odd hole">hasOddHole</a> -- tells whether a graph contains an odd hole</span></li>
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<div class="waystouse"><h2>Ways to use <tt>allOddHoles</tt> :</h2>
<ul><li>allOddHoles(Graph)</li>
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