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<head><title>numConnectedGraphComponents -- returns the number of connected components in a graph</title>
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<div><h1>numConnectedGraphComponents -- returns the number of connected components 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>d = numConnectedGraphComponents G</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>G</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>d</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the number of connected components of G</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function returns the number of connected components of a graph. A connected component of a graph is any maximal set of vertices which are pairwise connected by a path. Isolated vertices, which are those not appearing in any edge, count as connected components. This is in contrast to <a href="_num__Connected__Components.html" title="returns the number of connected components in a (hyper)graph">numConnectedComponents</a> in which isolated vertices are not counted as 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>
<p>The algorithm used by <tt>numConnectedGraphComponents</tt> turns <tt>G</tt> into a simplicial complex, and then computes the rank of the 0<sup>th</sup> reduced homology group. This number plus 1 plus the number of isolated vertices of <tt>G</tt> gives the number of connected components of <tt>G</tt>.</p>
<div>This method is intended to match the most common meaning for the number of connected components of a graph. This method can also be used on hypergraphs.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a..e];</pre>
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<tr><td><pre>i2 : g = graph {a*b,b*c,c*d,d*e,a*e} -- the 5-cycle (connected)
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o2 : Graph</pre>
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<tr><td><pre>i3 : h = graph {a*b,b*c,c*a,d*e} -- a 3-cycle and a disjoint edge (not connected)
o3 = Graph{edges => {{a, b}, {a, c}, {b, c}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o3 : Graph</pre>
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<tr><td><pre>i4 : k = graph {a*b,b*c,c*d,a*d} -- 4-cycle and isolated vertex (not connected)
o4 = Graph{edges => {{a, b}, {b, c}, {a, d}, {c, d}}}
ring => S
vertices => {a, b, c, d, e}
o4 : Graph</pre>
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<tr><td><pre>i5 : numConnectedGraphComponents g
o5 = 1</pre>
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<tr><td><pre>i6 : numConnectedGraphComponents h
o6 = 2</pre>
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<tr><td><pre>i7 : numConnectedGraphComponents k
o7 = 2</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__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>
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<div class="waystouse"><h2>Ways to use <tt>numConnectedGraphComponents</tt> :</h2>
<ul><li>numConnectedGraphComponents(HyperGraph)</li>
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