<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>hasOddHole -- tells whether a graph contains an odd hole</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Hyper__Graph.html">next</a> | <a href="_has__Good__Leaf.html">previous</a> | <a href="___Hyper__Graph.html">forward</a> | <a href="_has__Good__Leaf.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>hasOddHole -- tells whether a graph contains an odd hole</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>b = hasOddHole G</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>a <a href="___Graph.html">graph</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <tt>true</tt> if <tt>G</tt> has an odd hole and <tt>false</tt> otherwise</span></li> </ul> </div> </li> </ul> </div> <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> </td></tr> <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> </td></tr> <tr><td><pre>i3 : hasOddHole G o3 = true</pre> </td></tr> <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> </td></tr> <tr><td><pre>i5 : hasOddHole H o5 = false</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_all__Odd__Holes.html" title="returns all odd holes in a graph">allOddHoles</a> -- returns all odd holes in a graph</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>hasOddHole</tt> :</h2> <ul><li>hasOddHole(Graph)</li> </ul> </div> </div> </body> </html>