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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; Modeling it as a Graph Theory Problem</div>
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        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
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        <a class="toc_link" href="page4.xml">Objectives</a>
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        <a class="toc_link" href="">Foundations</a>
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        <a class="toc_link" href="page5.xml">What is a Graph?</a>
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        <a class="toc_link" href="page7.xml">Edges and Vertices</a>
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        <a class="toc_link" href="page10.xml">A First Theorem</a>
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        <a class="toc_link" href="page12.xml">Types of Graphs</a>
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        <a class="toc_link" href="">Solving the Drilling Problem</a>
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        <a class="toc_link" href="page24.xml">Independent Sets, Cliques, etc.</a>
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        <a class="toc_link" href="page26.xml">Vertex Coloring</a>
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        <a class="toc_link" href="page28.xml">Modeling it as a Graph Theory Problem</a>
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        <a class="toc_link" href="page31.xml">First Lower Bound</a>
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        <a class="toc_link" href="page33.xml">The Inside View</a>
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        <a class="toc_link" href="page38.xml">What's Next?</a>
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    <div id="content"><h1>Modeling it as a Graph Theory Problem</h1>
				<h2>Modeling The Drilling Problem As A Graph Theory Problem</h2>
				<p>
					It is clear that achieving both of our goals would require
					constructing large independent sets.
				</p>
				<p>
					It is easy to see that in our graph the largest independent
					set is of size 7.  This set is shown below; we will call it
					<m:math>
						<m:msub>
							<m:mi>I</m:mi>
							<m:mrow>
								<m:mi>l</m:mi>
								<m:mi>a</m:mi>
							</m:mrow>
						</m:msub>
					</m:math>.
				</p>
				<div class="box">

					
						<div class="figure" align="center"><a name=""/><img src="../graphics/plane7.gif" alt="The largest independent set."/><br/><b><i>Figure 27:
		The largest independent set.</i></b><br/><i>
								The only independent set of size 7.
							</i><br/><br/></div>
					
				</div>
				<p>
				Even though we claim it is supposed to be easy, don't take our word for it.
				Make sure it is <b>really</b> clear to you that this set 
				of 7 vertices is indeed the largest.
				</p>
				
					<p><div class="problem"><a name="problem29"/><b>Problem
				29:
			</b><br/>
						
							<p>
								Can you construct any independent sets of size 6?
							</p>
						
					</div></p>
					<p><div class="problem"><a name="problem30"/><b>Problem
				30:
			</b><br/>
						
						<p>
						Are there any independent sets of size 6 that are disjoint from
						<m:math>
							<m:msub>
								<m:mi>I</m:mi>
 								<m:mrow>
									<m:mi>l</m:mi>
									<m:mi>a</m:mi>
								</m:mrow>
							</m:msub>
						</m:math>?
						</p>
						
					</div></p>
				
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