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<head><title>connectedComponents -- returns the connected components of a hypergraph</title>
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<div><h1>connectedComponents -- returns the connected components of a 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>L = connectedComponents H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</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>, of lists of vertices. Each list of vertices is a connected component of H.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function returns the connected components of a hypergraph. A connected component of a hypergraph is any maximal set of vertices which are pairwise connected by a non-trivial path. Isolated vertices, which are those not appearing in any edge, do not appear in any connected components. This is in contrast to <a href="_connected__Graph__Components.html" title="returns the connected components of a graph">connectedGraphComponents</a> in which isolated vertices form their own connected components. See the <a href="___Connected_sp__Components_sp__Tutorial.html" title="clarifying the difference between graph and hypergraph components">Connected Components Tutorial</a> for more information.</p>
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<table class="examples"><tr><td><pre>i1 : R = QQ[a..l];</pre>
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<tr><td><pre>i2 : H = hyperGraph {a*b*c, c*d, d*e*f, h*i, i*j, l}
o2 = HyperGraph{edges => {{a, b, c}, {c, d}, {d, e, f}, {h, i}, {i, j}, {l}}}
ring => R
vertices => {a, b, c, d, e, f, g, h, i, j, k, l}
o2 : HyperGraph</pre>
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<tr><td><pre>i3 : L = connectedComponents H
o3 = {{a, b, c, d, e, f}, {h, i, j}, {l}}
o3 : List</pre>
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<tr><td><pre>i4 : apply(L, C -> inducedHyperGraph(H,C))
o4 = {HyperGraph{edges => {{a, b, c}, {c, d}, {d, e, f}}},
ring => QQ[a, b, c, d, e, f]
vertices => {a, b, c, d, e, f}
------------------------------------------------------------------------
HyperGraph{edges => {{h, i}, {i, j}}}, HyperGraph{edges => {{l}} }}
ring => QQ[h, i, j] ring => QQ[l]
vertices => {h, i, j} vertices => {l}
o4 : List</pre>
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<p>In the following example, hypergraph <tt>H</tt> contains the isolated vertex <tt>d</tt> and the vertex <tt>c</tt> which is in an edge of size one. Notice that <tt>d</tt> does not appear in any connected component while <tt>c</tt> does.</p>
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<table class="examples"><tr><td><pre>i5 : R = QQ[a,b,c,d];</pre>
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<tr><td><pre>i6 : H = hyperGraph {a*b, c}
o6 = HyperGraph{edges => {{a, b}, {c}} }
ring => R
vertices => {a, b, c, d}
o6 : HyperGraph</pre>
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<tr><td><pre>i7 : connectedComponents H
o7 = {{a, b}, {c}}
o7 : List</pre>
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<tr><td><pre>i8 : isolatedVertices H
o8 = {d}
o8 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Connected_sp__Components_sp__Tutorial.html" title="clarifying the difference between graph and hypergraph components">Connected Components Tutorial</a> -- clarifying the difference between graph and hypergraph components</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="_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="_isolated__Vertices.html" title="returns all vertices not contained in any edge">isolatedVertices</a> -- returns all vertices not contained in any edge</span></li>
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<div class="waystouse"><h2>Ways to use <tt>connectedComponents</tt> :</h2>
<ul><li>connectedComponents(HyperGraph)</li>
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