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    <div id="titlebar">Graph Theory: Industrial Drilling</div>
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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Foundations
					&gt; What is a Graph?</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="page38.xml">What's Next?</a>
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    <div id="content"><h1>What is a Graph?</h1>
				<p>
				A <i>graph</i> is represented as
				<m:math>
					<m:mi>G</m:mi>
					<m:mo>=</m:mo>
					<m:mfenced>
						<m:mi>V</m:mi>
						<m:mi>E</m:mi>
					</m:mfenced>
				</m:math>, where
				</p>
					<ol><li>
						<m:math><m:mi>V</m:mi></m:math>: a non-empty set of elements called <i>vertices</i>.
						</li><li>
						<m:math><m:mi>E</m:mi></m:math>: a set of pairs of vertices that comprise the <i>edges</i> of the graph.
						</li><li>
						<m:math><m:mo>|</m:mo><m:mi>V</m:mi><m:mo>|</m:mo></m:math> = number of vertices.
						</li><li>
						<m:math><m:mo>|</m:mo><m:mi>E</m:mi><m:mo>|</m:mo></m:math> = number of edges.
						</li></ol>
				<div class="box">
					<b><i>Example:</i></b>
					
						<div class="figure" align="center"><a name=""/><img src="../graphics/simplegraph1.gif" alt="A Graph"/><br/><b><i>Figure 4:
		A Graph</i></b><br/><i>
								A graph with
								<m:math>
									<m:mi>V</m:mi>
									<m:mo>=</m:mo>
									<m:mfenced close="}" open="{">
										<m:mi>a</m:mi>
										<m:mi>b</m:mi>
										<m:mi>c</m:mi>
										<m:mi>d</m:mi>
									</m:mfenced>
								</m:math> and
								<m:math>
									<m:mi>E</m:mi>
									<m:mo>=</m:mo>
									<m:mfenced close="}" open="{">
										<m:mfenced>
											<m:mi>a</m:mi>
											<m:mi>b</m:mi>
										</m:mfenced>
										<m:mfenced>
											<m:mi>a</m:mi>
											<m:mi>c</m:mi>
										</m:mfenced>
										<m:mfenced>
											<m:mi>b</m:mi>
											<m:mi>c</m:mi>
										</m:mfenced>
									</m:mfenced>
								</m:math>
								can be drawn as shown here.
							</i><br/><br/></div>
					
				</div>
				<p>
				If both <m:math><m:mo>|</m:mo><m:mi>V</m:mi><m:mo>|</m:mo></m:math> and
				<m:math><m:mo>|</m:mo><m:mi>E</m:mi><m:mo>|</m:mo></m:math> are finite,
				<m:math><m:mi>G</m:mi></m:math> is called <i>finite</i>.
				</p>
				<p>
					A graph with <m:math><m:mi>p</m:mi></m:math> vertices and
					<m:math><m:mi>q</m:mi></m:math> edges is called a
					<i>p,q-graph</i>.
				</p>
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