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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; Walks, Paths, Cycles</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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    <div id="content"><h1>Walks, Paths, Cycles</h1>
				<p>
				A <i>walk</i> from vertex <m:math><m:mi>v</m:mi></m:math> to a vertex <m:math><m:mi>u</m:mi></m:math> 
				is a sequence of alternating vertices and edges
				</p>
				<p><div class="equation" align="center">
					<m:math>
						<m:msub>
							<m:mi>v</m:mi>
							<m:mn>1</m:mn>
						</m:msub>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mn>1</m:mn>
						</m:msub>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>v</m:mi>
							<m:mn>2</m:mn>
						</m:msub>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mn>2</m:mn>
						</m:msub>
						<m:mo>,</m:mo>
						<m:mo>…</m:mo>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mrow>
								<m:mi>k</m:mi>
								<m:mo>-</m:mo>
								<m:mn>1</m:mn>
							</m:mrow>
						</m:msub>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>v</m:mi>
							<m:mi>k</m:mi>
						</m:msub>
						<m:mo>,</m:mo>
					</m:math>
				<br/><b><i>Equation
				6</i></b></div></p>
				<p>
				such that
				</p>
				<ol><li>
					<m:math>
						<m:msub>
							<m:mi>v</m:mi>
							<m:mn>1</m:mn>
						</m:msub>
						<m:mo>=</m:mo>
						<m:mi>v</m:mi>
					</m:math>, 
      				<m:math>
      					<m:msub>
							<m:mi>v</m:mi>
							<m:mi>k</m:mi>
						</m:msub>
						<m:mo>=</m:mo>
						<m:mi>u</m:mi>
      				</m:math>, and
					</li><li>
					<m:math>
						<m:mo>∀</m:mo>
						<m:mi>i</m:mi>
						<m:mo>=</m:mo>
						<m:mn>1</m:mn>
						<m:mo>,</m:mo>
						<m:mo>…</m:mo>
						<m:mo>,</m:mo>
						<m:mi>k</m:mi>
						<m:mo>-</m:mo>
						<m:mn>1</m:mn>
						<m:mo>,</m:mo>
						<m:msub>
							<m:mi>e</m:mi>
							<m:mi>i</m:mi>
						</m:msub>
					</m:math> 
      				is incident with <m:math><m:msub><m:mi>v</m:mi><m:mi>i</m:mi></m:msub></m:math> and
      				<m:math>
      					<m:msub>
      						<m:mi>v</m:mi>
      						<m:mrow>
      							<m:mi>i</m:mi>
      							<m:mo>+</m:mo>
      							<m:mn>1</m:mn>
      						</m:mrow>
      					</m:msub>
      				</m:math>.
					</li></ol>
				<p>
				A walk is called a <i>path</i> if it does not repeat any vertex.
				</p>
				<p>
				A graph is called <i>connected</i> if, for any two vertices <m:math><m:mi>u</m:mi></m:math> 
				and <m:math><m:mi>v</m:mi></m:math> of the graph, there is a walk from <m:math><m:mi>u</m:mi></m:math> to <m:math><m:mi>v</m:mi></m:math>.
				</p>
				<p>
				A <i>connected component</i> is a maximal connected subgraph. 
				A subset <m:math><m:mi>S</m:mi></m:math> of vertices of a graph <m:math><m:mi>G</m:mi></m:math> is a connected component if and only if:
				</p>
				<ol><li>
					For any two vertices in <m:math><m:mi>S</m:mi></m:math>, there is a path connecting them.
					</li><li>
					Any vertex not in <m:math><m:mi>S</m:mi></m:math> is not adjacent to any vertex in <m:math><m:mi>S</m:mi></m:math>.
					</li></ol>
				<p>
				A <i>closed walk</i> is a walk whose first and last vertices are 
				the same.
				</p>
				<p>
				A <i>cycle</i> is a closed walk which does not repeat any vertex, 
				except for the first and the last.
				</p>
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