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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="page15.xml">Subgraphs and Supergraphs</a>
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        <a class="toc_link" href="page18.xml">Walks, Paths, Cycles</a>
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        <a class="toc_link" href="page21.xml">Trees</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="page34.xml">A Good Coloring</a>
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        <a class="toc_link" href="page37.xml">The Best Coloring</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>
				Now that we have covered the fundamentals of graph theory, let us return to the 
				drilling problem. In order to use what we have learned about graph theory, we will reformulate 
				the problem as a <b>graph theory problem</b> as shown below:
				</p>
				<p>
				Recall the pattern that was to be drilled into the workpiece by the gang drill:
				</p>
				<div class="box">
					<b><i>The Final Product:</i></b>
					
						<div class="figure" align="center"><a name=""/><img src="../graphics/plane.gif" alt="The Final Product"/><br/><b><i>Figure 26:
		The Final Product</i></b><br/><i>
								The pattern of holes that had to be drilled into the
								workpiece with a gang drill.
							</i><br/><br/></div>
					
				</div>
				<p>
				The set of 91 points on the plane is viewed as a set of vertices in a graph <m:math><m:mi>G</m:mi></m:math>. 
				There are exactly 91 vertices. Each of them corresponds to a point; 
				different vertices correspond to different points.
				</p>
				<p>
				The set of edges in the graph <m:math><m:mi>G</m:mi></m:math> is the set of pairs of vertices 
				whose corresponding vertices are less than 35 units apart.
				</p>
				<p>
				Any set of points that can be drilled in one pass corresponds to 
				vertices in <m:math><m:mi>G</m:mi></m:math> that form an independent set.
				</p>
				<p>
				The sequence of passes that would yield all points drilled corresponds 
				to a set of independent sets that cover the whole graph. 
				Such a set of independent sets is a vertex coloring of the graph.
				</p>
				<div class="box">
					<b><i>Conclusion</i></b>
					
					<p>
					The drilling problem is identical to the problem of coloring the graph <m:math><m:mi>G</m:mi></m:math>
					using the smallest possible number of colors.
					Recall that this number is called the chromatic number of the graph.
					</p>
					
				</div>
				<p>
				We have two objectives:
				</p>
				<ol><li>
					Construct a coloring of the graph; and
					</li><li>
					Prove that the number of colors used is minimum.
					</li></ol>
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