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<head><title>inducedGraph -- returns the induced subgraph of a graph</title>
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<div><h1>inducedGraph -- returns the induced subgraph of 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>h = inducedGraph(G, L)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>a <a href="___Graph.html">graph</a></span></span></li>
<li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of vertices (i.e. variables in the ring of <tt>G</tt>)</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>h</tt>, <span>a <a href="___Graph.html">graph</a></span>, the induced subgraph of <tt>G</tt> whose edges are contained in <tt>L</tt></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_induced__Graph_lp..._cm_sp__Original__Ring_sp_eq_gt_sp..._rp.html">OriginalRing => ...</a>, -- use OriginalRing inside inducedGraph or inducedHyperGraph</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function returns the induced subgraph of a graph on a specified set of vertices. This function enables the user to create subgraphs of the original graph. <tt>inducedGraph</tt> accepts a <a href="___Graph.html" title="a class for graphs">Graph</a> as input and returns a <a href="___Graph.html" title="a class for graphs">Graph</a> as well. Use <a href="_induced__Hyper__Graph.html" title="returns the induced subgraph of a (hyper)graph">inducedHyperGraph</a> for a <a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a> instead.</p>
<p>The default option is for the ring of the induced subgraph to contain only variables in <tt>L</tt>. Then the current ring must be changed before working with the induced subgraph. We use this setup to avoid having a lot of isolated vertices in the induced graph. However, one can set the option <a href="___Original__Ring.html" title="optional argument for inducedHyperGraph">OriginalRing</a> to <tt>true</tt> if one wants to give the induced graph the same ring as the original graph.</p>
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<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c,d,e];</pre>
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<tr><td><pre>i2 : G = graph {a*b,b*c,c*d,d*e,e*a} -- graph of the 5-cycle
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o2 : Graph</pre>
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<tr><td><pre>i3 : H1 = inducedGraph(G,{b,c,d,e})
o3 = Graph{edges => {{b, c}, {c, d}, {d, e}}}
ring => QQ[b, c, d, e]
vertices => {b, c, d, e}
o3 : Graph</pre>
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<tr><td><pre>i4 : H2 = inducedGraph(G,{a,b,d,e})
o4 = Graph{edges => {{a, b}, {a, e}, {d, e}}}
ring => QQ[a, b, d, e]
vertices => {a, b, d, e}
o4 : Graph</pre>
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<tr><td><pre>i5 : use ring H1
o5 = QQ[b, c, d, e]
o5 : PolynomialRing</pre>
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<tr><td><pre>i6 : inducedGraph(H1,{c,d,e})
o6 = Graph{edges => {{c, d}, {d, e}}}
ring => QQ[c, d, e]
vertices => {c, d, e}
o6 : Graph</pre>
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<tr><td><pre>i7 : use ring G
o7 = R
o7 : PolynomialRing</pre>
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<tr><td><pre>i8 : inducedGraph(G,{b,c,d,e},OriginalRing=>true) --H1 but in bigger ring
o8 = Graph{edges => {{b, c}, {c, d}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o8 : Graph</pre>
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<div>Equivalently, one can use <a href="_change__Ring.html" title="replaces vertices with variables of a different ring">changeRing</a> (and <a href="_graph.html" title="constructor for Graph">graph</a>) to move the induced graph back into the original ring.</div>
<table class="examples"><tr><td><pre>i9 : R = QQ[a,b,c,d,e];</pre>
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<tr><td><pre>i10 : G = graph {a*b,b*c,c*d,d*e,e*a} -- graph of the 5-cycle
o10 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
ring => R
vertices => {a, b, c, d, e}
o10 : Graph</pre>
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<tr><td><pre>i11 : H = inducedGraph(G,{b,c,d})
o11 = Graph{edges => {{b, c}, {c, d}}}
ring => QQ[b, c, d]
vertices => {b, c, d}
o11 : Graph</pre>
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<tr><td><pre>i12 : graph changeRing(H,R,{b,c,d})
o12 = Graph{edges => {{d, c}, {c, b}} }
ring => R
vertices => {a, b, c, d, e}
o12 : Graph</pre>
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<div class="single"><h2>See also</h2>
<ul><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="_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="_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>
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<div class="waystouse"><h2>Ways to use <tt>inducedGraph</tt> :</h2>
<ul><li>inducedGraph(Graph,List)</li>
</ul>
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