<?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>EdgeIdeals -- A package for working with the edge ideals of (hyper)graphs</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_adjacency__Matrix.html">next</a> | previous | <a href="_adjacency__Matrix.html">forward</a> | backward | up | top | <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>EdgeIdeals -- A package for working with the edge ideals of (hyper)graphs</h1>
<div class="single"><h2>Description</h2>
<div><p><em>EdgeIdeals</em> is a package to work with the edge ideals of (hyper)graphs.</p>
<p>An edge ideal is a square-free monomial ideal where the generators of the monomial ideal correspond to the edges of the (hyper)graph. An edge ideal complements the Stanley-Reisner correspondence (see <a href="../../SimplicialComplexes/html/index.html" title="simplicial complexes">SimplicialComplexes</a>) by providing an alternative combinatorial interpretation of the monomial generators.</p>
<p>This package exploits the correspondence between square-free monomial ideals and the combinatorial objects, by using commutative algebra routines to derive information about (hyper)graphs. For some of the mathematical background on this material, see Chapter 6 of the textbook <i>Monomial Algebras</i> by R. Villarreal and the survey paper of T. Ha and A. Van Tuyl ("Resolutions of square-free monomial ideals via facet ideals: a survey," Contemporary Mathematics. 448 (2007) 91-117).</p>
<p>See the <a href="___Constructor_sp__Overview.html" title="a summary of the many ways of making graphs and hypergraphs">Constructor Overview</a> and the <a href="___Extended_sp__Example.html" title="an extended example using EdgeIdeals">Extended Example</a> for some illustrations of ways to use this package.</p>
<div><b>Note:</b> We require all hypergraphs to be clutters, which are hypergraphs in which no edge is a subset of another. If <i>H</i> is a hypergraph that is not a clutter, then the edge ideal of <i>H</i> is indistinguishable from the edge ideal of the clutter of minimal edges in <i>H</i>. (Edges of <i>H</i> that are supersets of other edges would not appear as minimal generators of the edge ideal of <i>H</i>.) The edge ideal of a hypergraph is similar to the facet ideal of a simplicial complex, as defined by S. Faridi in "The facet ideal of a simplicial complex," Manuscripta Mathematica 109, 159-174 (2002).</div>
</div>
</div>
<div class="single"><h2>Authors</h2>
<ul><li><div class="single"><a href="http://www.math.okstate.edu/~chris/">Chris Francisco</a><span> <<a href="mailto:chris@math.okstate.edu">chris@math.okstate.edu</a>></span></div>
</li>
<li><div class="single"><a href="http://www.mathstat.dal.ca/~handrew/">Andrew Hoefel</a><span> <<a href="mailto:handrew@mathstat.dal.ca">handrew@mathstat.dal.ca</a>></span></div>
</li>
<li><div class="single"><a href="http://flash.lakeheadu.ca/~avantuyl/">Adam Van Tuyl</a><span> <<a href="mailto:avantuyl@lakeheadu.ca">avantuyl@lakeheadu.ca</a>></span></div>
</li>
</ul>
</div>
<div class="single"><h2>Certification</h2>
<p>Version <b>1.0.0</b> of this package was accepted for publication in <a href="http://j-sag.org/Volume1/">volume 1</a> of the journal <a href="http://j-sag.org/">The Journal of Software for Algebra and Geometry: Macaulay2</a> on 2009-06-27, in the article <a href="http://j-sag.org/Volume1/jsag-1-2009.pdf">EdgeIdeals: a package for (hyper)graphs</a>. That version can be obtained <a href="http://j-sag.org/Volume1/EdgeIdeals.m2">from the journal</a> or from the <em>Macaulay2</em> source code repository, after installing <a href="http://subversion.tigris.org/">subversion</a>, with the following shell command:</p>
<pre> svn export -r 9342 svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/EdgeIdeals.m2</pre>
<p>The following command will display the log messages accompanying any changes to the file in the repository since publication.</p>
<pre> svn log -r 9343:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/EdgeIdeals.m2</pre>
<p>The following command will summarize the changes to the file in the repository since publication, in the format the program <tt>diff</tt> uses: lines starting with <tt>+</tt> have been added, and lines starting with <tt>-</tt> have been removed. (Changes to white space or end of line style will not be reported.)</p>
<pre> svn diff -x "-b --ignore-eol-style" -r 9342:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/EdgeIdeals.m2</pre>
<p>The differences between two releases in the repository mentioned in the log can be displayed by replacing <tt>9342:HEAD</tt> by the pair of release numbers separated by a colon.</p>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>1.0.0</b> of EdgeIdeals.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/EdgeIdeals.m2">EdgeIdeals.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Graph.html" title="a class for graphs">Graph</a> -- a class for graphs</span></li>
<li><span><a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a> -- a class for hypergraphs</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><a href="_adjacency__Matrix.html" title="returns the adjacency Matrix of a graph">adjacencyMatrix</a> -- returns the adjacency Matrix of a graph</span></li>
<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="_all__Odd__Holes.html" title="returns all odd holes in a graph">allOddHoles</a> -- returns all odd holes in a graph</span></li>
<li><span><a href="_anti__Cycle.html" title="returns a graph of an anticycle">antiCycle</a> -- returns a graph of an anticycle</span></li>
<li><span><a href="_change__Ring.html" title="replaces vertices with variables of a different ring">changeRing</a> -- replaces vertices with variables of a different ring</span></li>
<li><span><a href="_chromatic__Number.html" title="computes the chromatic number of a hypergraph">chromaticNumber</a> -- computes the chromatic number of a hypergraph</span></li>
<li><span><a href="_clique__Complex.html" title="returns the clique complex of a graph">cliqueComplex</a> -- returns the clique complex of a graph</span></li>
<li><span><a href="_clique__Number.html" title="computes the clique number of a graph">cliqueNumber</a> -- computes the clique number of a graph</span></li>
<li><span><a href="_complement__Graph.html" title="returns the complement of a graph or hypergraph">complementGraph</a> -- returns the complement of a graph or hypergraph</span></li>
<li><span><a href="_complete__Graph.html" title="returns a complete graph">completeGraph</a> -- returns a complete graph</span></li>
<li><span><a href="_complete__Multi__Partite.html" title="returns a complete multipartite graph">completeMultiPartite</a> -- returns a complete multipartite graph</span></li>
<li><span><a href="_connected__Components.html" title="returns the connected components of a hypergraph">connectedComponents</a> -- returns the connected components of a hypergraph</span></li>
<li><span><a href="_connected__Graph__Components.html" title="returns the connected components of a graph">connectedGraphComponents</a> -- returns the connected components of a graph</span></li>
<li><span><a href="_cover__Ideal.html" title="creates the cover ideal of a (hyper)graph">coverIdeal</a> -- creates the cover ideal of a (hyper)graph</span></li>
<li><span><a href="_cycle.html" title="returns a graph cycle">cycle</a> -- returns a graph cycle</span></li>
<li><span><a href="_degree__Vertex.html" title="returns the degree of a vertex">degreeVertex</a> -- returns the degree of a vertex</span></li>
<li><span><a href="_delete__Edges.html" title="returns the (hyper)graph with specified edges removed">deleteEdges</a> -- returns the (hyper)graph with specified edges removed</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="_edges.html" title="gets the edges of a (hyper)graph">edges</a> -- gets the edges of a (hyper)graph</span></li>
<li><span><a href="_get__Cliques.html" title="returns cliques in a graph">getCliques</a> -- returns cliques in a graph</span></li>
<li><span><a href="_get__Edge.html" title="gets the n-th edge in a (hyper)graph">getEdge</a> -- gets the n-th edge in a (hyper)graph</span></li>
<li><span><a href="_get__Edge__Index.html" title="finds the index of an edge in a HyperGraph">getEdgeIndex</a> -- finds the index of an edge in a HyperGraph</span></li>
<li><span><a href="_get__Good__Leaf.html" title="returns an edge that is a good leaf">getGoodLeaf</a> -- returns an edge that is a good leaf</span></li>
<li><span><a href="_get__Good__Leaf__Index.html" title="returns the index of an edge that is a good leaf">getGoodLeafIndex</a> -- returns the index of an edge that is a good leaf</span></li>
<li><span><a href="_get__Max__Cliques.html" title="returns maximal cliques in a graph">getMaxCliques</a> -- returns maximal cliques in a graph</span></li>
<li><span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></li>
<li><span><a href="_has__Good__Leaf.html" title="determines if a HyperGraph contains a good leaf">hasGoodLeaf</a> -- determines if a HyperGraph contains a good leaf</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>
<li><span><a href="_hyper__Graph.html" title="constructor for HyperGraph">hyperGraph</a> -- constructor for HyperGraph</span></li>
<li><span><a href="_hyper__Graph__To__Simplicial__Complex.html" title="makes a simplicial complex from a (hyper)graph">hyperGraphToSimplicialComplex</a> -- makes a simplicial complex from a (hyper)graph</span></li>
<li><span><a href="_incidence__Matrix.html" title="returns the incidence matrix of a hypergraph">incidenceMatrix</a> -- returns the incidence matrix of a hypergraph</span></li>
<li><span><a href="_independence__Complex.html" title="returns the independence complex of a (hyper)graph">independenceComplex</a> -- returns the independence complex of a (hyper)graph</span></li>
<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>
<li><span><a href="_induced__Graph.html" title="returns the induced subgraph of a graph">inducedGraph</a> -- returns the induced subgraph of a graph</span></li>
<li><span><a href="_induced__Hyper__Graph.html" title="returns the induced subgraph of a (hyper)graph">inducedHyperGraph</a> -- returns the induced subgraph of a (hyper)graph</span></li>
<li><span><a href="_is__Bipartite.html" title="determines if a graph is bipartite">isBipartite</a> -- determines if a graph is bipartite</span></li>
<li><span><a href="_is__Chordal.html" title="determines if a graph is chordal">isChordal</a> -- determines if a graph is chordal</span></li>
<li><span><a href="_is__C__M.html" title="determines if a (hyper)graph is Cohen-Macaulay">isCM</a> -- determines if a (hyper)graph is Cohen-Macaulay</span></li>
<li><span><a href="_is__Connected.html" title="determines if a (hyper)graph is connected">isConnected</a> -- determines if a (hyper)graph is connected</span></li>
<li><span><a href="_is__Connected__Graph.html" title="determines if a graph is connected">isConnectedGraph</a> -- determines if a graph is connected</span></li>
<li><span><a href="_is__Edge.html" title="determines if an edge is in a (hyper)graph">isEdge</a> -- determines if an edge is in a (hyper)graph</span></li>
<li><span><a href="_is__Forest.html" title="determines whether a (hyper)graph is a forest">isForest</a> -- determines whether a (hyper)graph is a forest</span></li>
<li><span><a href="_is__Good__Leaf.html" title="determines if an edge is a good leaf">isGoodLeaf</a> -- determines if an edge is a good leaf</span></li>
<li><span><a href="_is__Graph.html" title="determines if a hypergraph is a graph">isGraph</a> -- determines if a hypergraph is a graph</span></li>
<li><span><a href="_is__Leaf.html" title="determines if an edge (or vertex) is a leaf of a (hyper)graph">isLeaf</a> -- determines if an edge (or vertex) is a leaf of a (hyper)graph</span></li>
<li><span><a href="_isolated__Vertices.html" title="returns all vertices not contained in any edge">isolatedVertices</a> -- returns all vertices not contained in any edge</span></li>
<li><span><a href="_is__Perfect.html" title="determines whether a graph is perfect">isPerfect</a> -- determines whether a graph is perfect</span></li>
<li><span><a href="_is__S__C__M.html" title="determines if a (hyper)graph is sequentially Cohen-Macaulay">isSCM</a> -- determines if a (hyper)graph is sequentially Cohen-Macaulay</span></li>
<li><span><a href="_line__Graph.html" title="returns the line graph of a (hyper)graph">lineGraph</a> -- returns the line graph of a (hyper)graph</span></li>
<li><span><a href="_neighbors.html" title="returns the neighbors of a vertex or list of vertices">neighbors</a> -- returns the neighbors of a vertex or list of vertices</span></li>
<li><span><a href="_num__Connected__Components.html" title="returns the number of connected components in a (hyper)graph">numConnectedComponents</a> -- returns the number of connected components in a (hyper)graph</span></li>
<li><span><a href="_num__Connected__Graph__Components.html" title="returns the number of connected components in a graph">numConnectedGraphComponents</a> -- returns the number of connected components in a graph</span></li>
<li><span><a href="_num__Triangles.html" title="returns the number of triangles in a graph">numTriangles</a> -- returns the number of triangles in a graph</span></li>
<li><span><a href="_random__Graph.html" title="returns a random graph">randomGraph</a> -- returns a random graph</span></li>
<li><span><a href="_random__Hyper__Graph.html" title="returns a random hypergraph">randomHyperGraph</a> -- returns a random hypergraph</span></li>
<li><span><a href="_random__Uniform__Hyper__Graph.html" title="returns a random uniform hypergraph">randomUniformHyperGraph</a> -- returns a random uniform hypergraph</span></li>
<li><span><a href="_simplicial__Complex__To__Hyper__Graph.html" title="makes a (hyper)graph from a simplicial complex">simplicialComplexToHyperGraph</a> -- makes a (hyper)graph from a simplicial complex</span></li>
<li><span><a href="_smallest__Cycle__Size.html" title="returns the size of the smallest induced cycle of a graph">smallestCycleSize</a> -- returns the size of the smallest induced cycle of a graph</span></li>
<li><span><a href="_spanning__Tree.html" title="returns a spanning tree of a graph">spanningTree</a> -- returns a spanning tree of a 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>
<li><span><a href="_vertices.html" title="gets the vertices of a (hyper)graph">vertices</a> -- gets the vertices of a (hyper)graph</span></li>
</ul>
</div>
</li>
<li><div class="single">Methods<ul><li><span>changeRing(HyperGraph,PolynomialRing,List), see <span><a href="_change__Ring.html" title="replaces vertices with variables of a different ring">changeRing</a> -- replaces vertices with variables of a different ring</span></span></li>
<li><span>degreeVertex(HyperGraph,RingElement), see <span><a href="_degree__Vertex.html" title="returns the degree of a vertex">degreeVertex</a> -- returns the degree of a vertex</span></span></li>
<li><span>degreeVertex(HyperGraph,ZZ), see <span><a href="_degree__Vertex.html" title="returns the degree of a vertex">degreeVertex</a> -- returns the degree of a vertex</span></span></li>
<li><span>deleteEdges(HyperGraph,List), see <span><a href="_delete__Edges.html" title="returns the (hyper)graph with specified edges removed">deleteEdges</a> -- returns the (hyper)graph with specified edges removed</span></span></li>
<li><span>getCliques(Graph,ZZ), see <span><a href="_get__Cliques.html" title="returns cliques in a graph">getCliques</a> -- returns cliques in a graph</span></span></li>
<li><span>getEdge(HyperGraph,ZZ), see <span><a href="_get__Edge.html" title="gets the n-th edge in a (hyper)graph">getEdge</a> -- gets the n-th edge in a (hyper)graph</span></span></li>
<li><span>getEdgeIndex(HyperGraph,List), see <span><a href="_get__Edge__Index.html" title="finds the index of an edge in a HyperGraph">getEdgeIndex</a> -- finds the index of an edge in a HyperGraph</span></span></li>
<li><span>getEdgeIndex(HyperGraph,RingElement), see <span><a href="_get__Edge__Index.html" title="finds the index of an edge in a HyperGraph">getEdgeIndex</a> -- finds the index of an edge in a HyperGraph</span></span></li>
<li><span><a href="___Hyper__Graph_sp_eq_eq_sp__Hyper__Graph.html" title="equality">HyperGraph == HyperGraph</a> -- equality</span></li>
<li><span>inducedGraph(Graph,List), see <span><a href="_induced__Graph.html" title="returns the induced subgraph of a graph">inducedGraph</a> -- returns the induced subgraph of a graph</span></span></li>
<li><span>inducedHyperGraph(HyperGraph,List), see <span><a href="_induced__Hyper__Graph.html" title="returns the induced subgraph of a (hyper)graph">inducedHyperGraph</a> -- returns the induced subgraph of a (hyper)graph</span></span></li>
<li><span>isEdge(HyperGraph,List), see <span><a href="_is__Edge.html" title="determines if an edge is in a (hyper)graph">isEdge</a> -- determines if an edge is in a (hyper)graph</span></span></li>
<li><span>isEdge(HyperGraph,RingElement), see <span><a href="_is__Edge.html" title="determines if an edge is in a (hyper)graph">isEdge</a> -- determines if an edge is in a (hyper)graph</span></span></li>
<li><span>isGoodLeaf(HyperGraph,ZZ), see <span><a href="_is__Good__Leaf.html" title="determines if an edge is a good leaf">isGoodLeaf</a> -- determines if an edge is a good leaf</span></span></li>
<li><span>isLeaf(Graph,ZZ), see <span><a href="_is__Leaf.html" title="determines if an edge (or vertex) is a leaf of a (hyper)graph">isLeaf</a> -- determines if an edge (or vertex) is a leaf of a (hyper)graph</span></span></li>
<li><span>isLeaf(HyperGraph,RingElement), see <span><a href="_is__Leaf.html" title="determines if an edge (or vertex) is a leaf of a (hyper)graph">isLeaf</a> -- determines if an edge (or vertex) is a leaf of a (hyper)graph</span></span></li>
<li><span>isLeaf(HyperGraph,ZZ), see <span><a href="_is__Leaf.html" title="determines if an edge (or vertex) is a leaf of a (hyper)graph">isLeaf</a> -- determines if an edge (or vertex) is a leaf of a (hyper)graph</span></span></li>
<li><span>neighbors(HyperGraph,List), see <span><a href="_neighbors.html" title="returns the neighbors of a vertex or list of vertices">neighbors</a> -- returns the neighbors of a vertex or list of vertices</span></span></li>
<li><span>neighbors(HyperGraph,RingElement), see <span><a href="_neighbors.html" title="returns the neighbors of a vertex or list of vertices">neighbors</a> -- returns the neighbors of a vertex or list of vertices</span></span></li>
<li><span>neighbors(HyperGraph,ZZ), see <span><a href="_neighbors.html" title="returns the neighbors of a vertex or list of vertices">neighbors</a> -- returns the neighbors of a vertex or list of vertices</span></span></li>
</ul>
</div>
</li>
<li><div class="single">Symbols<ul><li><span><a href="___Branch__Limit.html" title="optional argument for randomHyperGraph">BranchLimit</a> -- optional argument for randomHyperGraph</span></li>
<li><span><a href="___Gins.html" title="optional argument for isSCM">Gins</a> -- optional argument for isSCM</span></li>
<li><span><a href="___Maximal__Edges.html" title="optional argument for changeRing">MaximalEdges</a> -- optional argument for changeRing</span></li>
<li><span><a href="___Original__Ring.html" title="optional argument for inducedHyperGraph">OriginalRing</a> -- optional argument for inducedHyperGraph</span></li>
<li><span><a href="___Time__Limit.html" title="optional argument for randomHyperGraph">TimeLimit</a> -- optional argument for randomHyperGraph</span></li>
</ul>
</div>
</li>
</ul>
</div>
</div>
</body>
</html>
