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<div><h1>Connected Components Tutorial -- clarifying the difference between graph and hypergraph components</h1>
<div><p>In this tutorial, we discuss the various methods that deal with connected components of graphs and hypergraphs. Our main objective is to make a distinction between the two different definitions of connected components that are used in the <a href="index.html" title="A package for working with the edge ideals of (hyper)graphs">EdgeIdeals</a> package.</p>
<p>A vertex of a (hyper)graph <tt>H</tt> said to be an isolated vertex if it is not contained in any edge of <tt>H</tt>. In particular, if a vertex of <tt>H</tt> is contained in a edge of size one then it is not considered isolated.</p>
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<table class="examples"><tr><td><pre>i1 : R = QQ[u,v,x,y,z];</pre>
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<tr><td><pre>i2 : H = hyperGraph({{u,v},{x}});</pre>
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<tr><td><pre>i3 : isolatedVertices H
o3 = {y, z}
o3 : List</pre>
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<p><em>Graph Components</em>. A connected component of a graph is any maximal set of vertices which are pairwise connected by a (possibly trivial) path. The most important part of this definition is that isolated vertices count as connected components.</p>
<p>The following methods use this definition of a connected component: <a href="_connected__Graph__Components.html" title="returns the connected components of a graph">connectedGraphComponents</a>, <a href="_num__Connected__Graph__Components.html" title="returns the number of connected components in a graph">numConnectedGraphComponents</a> and <a href="_is__Connected__Graph.html" title="determines if a graph is connected">isConnectedGraph</a>.</p>
<p><em>Hypergraph Components</em>. A connected component of a hypergraph is any maximal set of vertices which are pairwise connected by a non-trivial path. Here isolated vertices do not count as connected components.</p>
<p>The following methods use the hypergraph definition of a connected component: <a href="_connected__Components.html" title="returns the connected components of a hypergraph">connectedComponents</a>, <a href="_num__Connected__Components.html" title="returns the number of connected components in a (hyper)graph">numConnectedComponents</a> and <a href="_is__Connected.html" title="determines if a (hyper)graph is connected">isConnected</a>.</p>
<p>The next example uses all of these methods on a graph to illustrate the difference between the two definitions.</p>
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<table class="examples"><tr><td><pre>i4 : R = QQ[u,v,x,y,z];</pre>
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<tr><td><pre>i5 : G = graph({{x,y},{y,z}});</pre>
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<tr><td><pre>i6 : isolatedVertices G
o6 = {u, v}
o6 : List</pre>
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<tr><td><pre>i7 : connectedGraphComponents G
o7 = {{u}, {v}, {x, y, z}}
o7 : List</pre>
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<tr><td><pre>i8 : numConnectedGraphComponents G
o8 = 3</pre>
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<tr><td><pre>i9 : isConnectedGraph G
o9 = false</pre>
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<tr><td><pre>i10 : connectedComponents G
o10 = {{x, y, z}}
o10 : List</pre>
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<tr><td><pre>i11 : numConnectedComponents G
o11 = 1</pre>
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<tr><td><pre>i12 : isConnected G
o12 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><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="_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="_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="_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>
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